^1 


IN  MEMORIAM 
FLORIAN  CAJORl 


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By  D*  A*  MQRRAY,  Ph.D., 

Formerly  Instructor  in  Matiikmatics  in  Cornell 

University  ;  Professor  of  Mathematics  in 

Daliiousie  College,  Halifax,  N.9. 


INTRODUCTORY  COURSE  IN  DIFFERENTIAL 
EQUATIONS,  FOR  Students  in  Classical  and 
Engineering  Colleges.     Pp.  xvi  +  2o6. 

PLANE  TRIGONOMETRY,  for  Colleges  and 
Secondary  Schools.  With  a  Protractor.  I'p.  xiii 
+  206. 

SPHERICAL  TRIGONOMETRY,  for  Colleges 
AND  Secondary  Schools.     Pp.  xiv  +  114. 

PLANE  AND  SPHERICAL  'J  RIGONOMETRY. 
In  One  Volume.     With  a  Protractor. 

LOGARITHMIC  AND  TRIGONOMETRIC 
TABLES.    Five-place  and  Four-place.   Pp.  99. 

PLANE    TRIGONOMETRY    AND    TABLES.      In 

One  Volume.     With  a  Protractor.     Pp.  318. 

A  FIRST  COURSE  IN  INFINITESIMAL  CAL- 
CULUS.    Pp.  xvii  +  439. 


NEW  YORK:  LONGMANS,  GREEN,  &  CO. 


A  FIRST  COURSE 


INFINITESIMAL    CALCULUS 


BY 


DANIEL    A.     MURRAY,   Ph.D.  (Johns  Hopkins) 
r 

PROFESSOR  OF  MATHEMATICS   IN  DALHOUSIE   COLLEGE, 
HALIFAX,  N.S. 


LONGMANS,    GREEN,    AND    CO. 

91  AND  93  FIFTH  AVENUE,  NEW  YORK 

LONDON  AND  BOMBAY 

1903 


Copyright,  1903,  by 
LONGMANS,    GREEN,   AND   CO. 


All  rights  reserved. 


J.  S.  Cashing  &  Co.  —  Berwick  ^  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE. 


This  book  has  been  written  for  beginners  in  calculus.  Its 
purpose  is  to  provide  an  introductory  course  for  those  who  are 
entering  upon  that  study  either  to  prepare  themselves  for  ele- 
mentary work  in  applied  science  or  to  gratify  and  develop  their 
interest  in  mathematics.  This  purpose  has  determined  the  choice 
and  the  arrangement  of  the  topics  and  the  mode  of  presentation. 
Little  more  has  been  discussed  than  what  may  be  regarded  as  the 
essentials  of  a  primary  course  in  calculus.  An  attempt  is  made 
to  describe  and  emphasise  the  fundamental  principles  of  the 
subject  in  such  a  way  that,  as  much  as  may  reasonably  be  ex- 
pected, they  may  be  clearly  understood,  firmly  grasped,  and 
intelligently  applied  by  young  students.  There  has  also  been 
kept  in  view  the  development  in  them  of  the  ability  to  read 
mathematics  and  to  prosecute  its  study  by  themselves. 

Excepting  in  a  few  instances,  only  real  functions  of  real 
variables  are  considered.  Simple,  practical  applications  of  the 
more  elementary  notions  are  introduced  as  early  as  possible; 
and,  subject  to  the  requirements  of  a  logically  counected  develop- 
ment of  the  study,  the  more  difficult  and  abstract  discussions 
appear  later.  In  accordance  with  this  plan,  the  time-honoured 
division  into  differential  calculus  and  integral  calculus  has  not 
been  made,  and,  to  mention  one  instance  in  particular,  following 
the  example  set  by  Professors  Lamb,  Gibson,  and  others,  the 
development  of  functions  in  series  is  taken  up  in  the  latter, 
instead  of  in  the  earlier,  part  of  the  course.  The  book,  however, 
can  be  divided  easily  into  differential  and  integral  sections,  and 
thus  can  be  adapted,  in  this  respect  at  least,  for  use  in  cases  in 
which  such  a  division  is  deemed  necessary. 

With  regard  to  simplicity  and  clearness  in  the  exposition  of 
the  subject,  it  may  be  said  that  the  aim  has  been  to  write  a  book 


vi  PREFACE. 

that  will  be  found  helpful  by  those  who  begin  the  study  of 
calculus  without  the  guidance  and  aid  of  a  teacher.  For  these 
students  more  especially,  throughout  the  work  suggestions  and 
remarks  are  made  concerning  the  order  in  which  the  various 
topics  may  be  studied,  the  relative  importance  of  the  various 
topics  in  a  first  study  of  calculus,  the  articles  that  must  be 
thoroughly  mastered,  and  the  articles  that  may  advantageously 
be  omitted  or  lightly  passed  over  at  the  first  reading,  and  so  on. 

The  notion  of  anti-differentiation  is  presented  simultaneously 
with  the  notion  of  differentiation,  and  exercises  thereon  appear 
early  in  the  text;  but  in  the  chapter  in  which  integration  is 
formally  taken  up  the  idea  of  integration  as  a  process  of  summa- 
tion is  considered  before  the  idea  of  integration  as  a  process 
which  is  the  inverse  of  differentiation.  In  this  matter  I  have 
followed  the  order  adopted  in  my  Integral  Calculus,  although 
there  is  considerable  difference  of  opinion  as  to  the  propriety 
or  the  advantage  of  this  order.  The  decision  to  follow  it  here 
has  been  made  mainly  for  the  reason  that  students  appear  —  at 
least  so  it  seems  to  me,  but  other  teachers  may  have  a  different 
experience  —  to  understand  more  clearly  and  vividly  the  relation 
of  integration  to  many  practical  problems  when  the  summation 
idea  is  put  in  the  forefront.  In  teaching  the 'one  order  can  be 
taken  as  readily  as  the  other. 

In  several  technical  schools  the  time  assigned  to  calculus  is 
not  sufficient  for  a  fair  study  of  Taylor's  theorem.  What  may 
be  regarded  as  the  irreducible  requisite  for  a  slight  working 
acquaintance  with  Taylor's  and  Maclaurin's  series  is  indicated 
at  the  beginning  of  Chapter  XIX.,  and  may  be  taken  at  an  early 
stage  in  the  course. 

The  evaluation  of  indeterminate  forms,  which  affords  interest- 
ing exercises  in  the  application  of  differentiation,  is  far  from 
being  as  important  as  many  other  applications  of  the  calculus; 
and  in  the  few  cases  in  which  this  evaluation  is  required  it  can 
be  effected  by  other  means.  Useful  exercises  in  applying  inte- 
gration can  be  given  to  students  who  have  a  knowledge  of 
mechanics.  In  many  cases,  however,  these  students  make  but 
a  small  fraction  of  the  class,  and,  besides,  in  a  large  number  of 
technical  schools  the  curriculum  provides  that  mechanics  shall 


PREFACE.  Vll 

follow  calculus.  Accordingly,  it  seemed  better  not  to  treat  inde- 
terminate forms  and  mechanics  in  the  body  of  the  text,  but  to 
deal  briefly  with  them  in  the  appendix. 

An  explanation  of  hyperbolic  functions  can  be  made  more 
naturally  and  more  fully,  perhaps,  in  a  course  in  calculus  than 
in  any  other  course  in  elementary  mathematics.  For  this  reason, 
and  also  because  students  will  meet  them  in  their  later  work 
and  reading,  a  note  on  these  functions  appears  in  the  latter  part 
of  the  book. 

Owing  to  the  pressure  of  other  subjects  the  time  allotted  to 
mathematics  in  quite  a  number  of  technical  schools  is  rather 
brief.  Where  this  is  the  case,  and  where  there  is  a  lack  of 
maturity  in  the  students,  it  is  better  not  to  try  to  cover  too 
much  ground,  but  to  lay  stress  on  fundamental  principles,  to 
drill  in  the  elementary  processes,  and  to  train  in  making  simple 
applications.  Thus  this  book,  small  as  it  may  be  regarded  even 
for  a  short  course,  contains  more  matter  than  can  be  thoroughly 
studied  in  the  few  months  allotted  to  calculus  in  colleges  and 
technical  schools  where  such  conditions  exist.  Several  topics, 
however  (for  example,  the  investigation  of  series),  which  in  some 
cases  are  not  studied  by  technical  students  owing  to  lack  of  time, 
are  very  important,  particularly  for  those  who  take  a  first  course 
in  the  calculus  as  an  introduction  to  a  more  extended  study  of 
the  subject  and  as  j^art  of  the  preparation  necessary  for  more 
advanced  work  in  mathematics.  For  the  sake  of  these  students 
more  especially,  but  not  exclusively  on  their  account,  many  definite 
references  for  collateral  reading  or  inspection  are  given  throughout 
the  text. 

It  is  hoped  that  these  references  will  add  to  the  helpfulness 
of  the  book.  With  but  very  few  exceptions  those  are  chosen 
which  are  easily  accessible  to  all  college  students.  Some  of 
the  references  will  aid  the  learner  by  presenting  an  idea  of  the 
text  in  the  words  of  another ;  but  the  larger  number  of  them 
are  intended  to  direct  students  to  places  where  they  will 
either  receive  fuller  information  or  be  impressed  with  some  of 
the  important  modern  ideas  of  mathematics.  Turning  up  such 
references  as  these  will  increase  the  mathematical  interest  of 
the  student  and  widen  his  outlook.     It  will  also  help  to  train 


viii  PREFACE. 

the  pupils  in  the  use  of  mathematical  literature,  and,  by  arous- 
ing and  exercising  their  critical  faculties,  will  greatly  benefit 
those  who  may  intend  to  teach  mathematics  in  the  secondary 
schools.  Of  course  the  lists  of  references  are  not  exhaustive, 
and,  while  care  has  been  taken  in  making  them,  it  is  to  be 
expected  that  several  other  equally  serviceable  lists  can  be 
arranged.  It  is  intended  that  these  lists  shall  be  revised  and 
supplemented  by  those  who  may  use  the  book. 

For  learners  who  can  afford  but  a  minimum  of  time  for  this 
study  the  essential  articles  of  a  short  course  are  indicated  after 
the  table  of  contents. 

The  exposition  given  here  is,  in  the  main,  a  result  of  my 
experience  in  teaching  the  calculus  to  a  large  number  of  pupils. 
Accordingly,  it  is  my  duty  to  acknowledge  my  indebtedness  to 
many  students  whose  difficulties  and  original  opinions  have 
interested  and  stimulated  me.  In  preparing  the  text  many 
works  and  articles  have  been  consulted.  I  feel  myself  to  be 
especially  indebted  to  the  writers  to  whom  references  are  made 
in  various  places  in  the  book. 

Not  many  examples  involving  a  technical  knowledge  of  engi- 
neering, physics,  or  chemistry  have  been  inserted.  Few  young 
students  understand  examples  of  this  kind  without  considerable 
explanation,  and  thus  it  seems  better  to  refer  the  pupils  to  the 
more  specialised  text-books  dealing  with  calculus  (for  instance, 
those  of  Perry,  Young  and  Linebarger,  and  Mellor),  which  contain 
many  examples  of  a  technical  character. 

I  take  this  opportunity  of  thanking  A.  T.  Bruegel,  M.M.E., 
Professor  of  Mechanical  Engineering  in  the  Drexel  Institute, 
Philadelphia,  for  advice  and  suggestions  concerning  the  draw- 
ings, and  Louis  C.  Loewenstein,  Ph.D.,  Instructor  in  Mechanical 
Engineering  in  Lehigh  University,  for  the  interest  and  care  taken 
by  him  in  making  the  figures.  I  also  wish  to  thank  Miss  A.  A. 
Stewart,  B.Sc,  and  my  colleague.  Professor  H.  Murray  for  kindly 
help  in  the  revision  of  the  proof-sheets.  Miss  Stewart  also  gave 
valuable  assistance  in  verifying  many  of  the  examples. 

D.  A.  MURRAY. 

Dalhousie  College,  Halifax,  N.S. 
August  15,  1903. 


CONTENTS. 


CHAPTER   I. 
Introductory  Problems. 

ART.  PAGE 


2.  Speed  of  a  moving  train 

3.  To  determine  the  speed  of  a  falling  body 

4.  To  determine  the  slope  of  a  tangent 

5.  To  determine  the  area  of  a  plane  figure   . 

6.  To  find  a  function  when  its  rate  of  change  is  known 
To  find  the  equation  of  a  curve  when  its  slope  is  known 

7.  Elementary  notions  used  in  infinitesimal  calculus   . 


2 
2 
6 
10 
11 
11 
11 


CHAPTER   XL 
Algebraic  Notions  which  are  frequently  used  in  the  Calculus. 

8.  Variables 13 

9.  Functions ,      .         .15 

10.  Constants 16 

11.  Classification  of  functions 17 

12.  Notation 17 

13.  Geometrical  representation  of  functions  of  one  variable  ...  19 

14.  Limits 20 

15.  Notation 24 

16.  Continuous  functions.     Discontinuous  functions     .         .         .         .    ■   24 

CHAPTER   III. 

Infinitesimals,  Derivatives,  Differentials,  Anti-derivatives,  and 
Anti-differentials. 

18.  Infinitesimals,  infinite  numbers,  finite  numbers 30 

19.  Orders  of  magnitude.    Orders  of  infinitesimals.    Orders  of  infinites  31 

20.  Theorems  on  limits  and  infinitesimals 35 

21.  Fundamental  theorems  of  the  calculus 36 

ix 


X  CONTENTS. 

ART.  PAGE 

22.  The  derivative  of  a  function  of  one  variable     ,         ....       38 

23.  Notation 41 

24.  The  geometrical  meaning  and  representation  of  the  derivative  of  a 

function 43 

25.  The  physical  meaning  of  the  derivative  of  a  function       ...  45 

26.  General  meaning  of  the  derivative  :  the  derivative  is  a  rate     .         .  46 

27.  Differentials 47 

27a.   Anti-derivatives  and  anti-differentials 50 

CHAPTER   IV. 
Differentiation  of  the  Ordinary  Functions. 

General  Besults  in  Differentiation. 

29.  The   derivative   of  the   sum   of   a  function  and  a  constant,  say 

0(x)  +  c 51 

30.  The  derivative  of  the  product  of  a  constant  and  a  function,  say 

C(/)(x) 53 

31.  The  derivative  of  the  sura  of  a  finite  number  of  functions        .         .  54 

32.  The  derivative  of  the  product  of  two  or  more  functions  ...  55 

33.  The  derivative  of  the  quotient  of  two  functions        ....  57 

34.  The  derivative  of  a  function  of  a  function 59 

35.  The  derivative  of  one  variable  with  respect  to  another  when  both 

are  functions  of  a  third  variable 60 

36.  Differentiation  of  inverse  functions 61 

Differentiation  of  Particular  Functions. 
A.    Algebraic  Functions. 

37.  Differentiation  of  m» 61 

B.   Lof/arithmic  and  Exponential  Functions. 

38.  N'oTE.    Tofind.lim^=oo(  1  + --)'" 66 

39-41.    Differentiation  of  log„  w,  a",  w" 67-71 

C.    Trigonometric  Functions. 
42-48.   Differentiation  of  sin  u.,  cos  m,  tan  i«,  cot  m,  sec  w,  esc  m,  vers  u     71-76 

D.   Inverse  Trigonometric  Functions. 

49-55.    Differentiation    of    sin"^  u,    cos~i  u^    tan"^  i«,    cot~i  ?/,    sec~^  w, 

csc-i  w,  vers-i  u 76-80 

56.   Differention  of  implicit  functions :  two  variables     ....      80 


CONTENTS.  XI 


CHAPTER  V. 


Some  Geometrical,  Physical,  and  Analytical  Applications. 
Geometric  Derivatives  and  Differentials. 

ART.  PAGE 

58.    Slope  of  a  curve  at  any  point :  rectangular  coordinates    ...  84 
69.    Lengths  of  tangent,  subtangent,  normal,  and  subnormal :  rectangu- 
lar coordinates 84 

60.  Slope  of  a  curve  at  any  point :  polar  coordinates      ....  88 

61.  Lengths   of  tangent,  subtangent,  normal,  and   subnormal:   polar 

coordinates 89 

62.  Applications  involving  rates 91 

63.  Rolle's  theorem 93 

64.  Theorem  of  mean  value 94 

65.  Small  errors  and  corrections  ;  relative  eiTor 96 

66.  Applications  to  algebra 98 

67.  Geometric  derivatives  and  differentials 99-106 


CHAPTER   VI. 
Successive  Differentiation. 

68.  Successive  derivatives 107 

69.  The  7ith  derivative  of  some  particular  functions       .         .        •,         .111 

70.  Successive  differentials 112 

71.  Successive  derivatives  of  y  with  respect  to  x  when  both  are  func- 

tions of  a  third  variable         112 

72.  Leibnitz's  theorem    . 113 

73.  Application  of  differentiation  to  elimination 114 


CHAPTER   VIL 
Further  Analytical  and  Geometrical  Applications. 

74.  Increasing  and  decreasing  functions 116 

75.  Maximum  and  minimum  values  of  a  function.     Critical  points  on 

the  graph,  and  critical  values  of  the  variable        .         .         .         .117 

76.  Inspection  of  the  critical  values  of  the  variable  for  maximum  or 

minimum  values  of  the  function 120 

77.  Practical  problems  in  maxima  and  minima      .....     123 

78.  Points  of  inflexion :  rectangular  coordinates    .         ,        .         .         .     127 


xu 


CONTENTS. 


CHAPTER   VIIL 


variables 


Differentiation  of  Functions  of  Several  Variables. 

ART. 

79.  Partial  derivatives.     Notation  .... 

80.  Successive  partial  derivatives    .... 

81.  Total  rate  of  variation  of  a  function  of  two  or  more 

82.  Total  differential 

83.  Approximate  value  of  small  errors   . 

84.  Differentiation  of  implicit  functions ;  two  variables 

85.  Order  of  partial  differentiations  commutative  . 

86.  Condition  that  an  expression  of  the  form  Pdx  + 

differential 

87.  Euler's  theorem  on  homogeneous  functions 

88.  Successive  total  derivatives       .... 


Qdy  be  a 


total 


PAGE 

130 
133 
134 
136 
138 
139 
140 

141 
142 
143 


CHAPTER   IX. 
Change  of  Variable. 

80.  Change  of  variable 144 

90.  Interchange  of  the  dependent  and  independent  variables  .        .  144 

91.  Change  of  the  dependent  variable 145 

92.  Change  of  the  independent  variable 145 

93.  Dependent  and  independent  variables  both  expressed  in  terms  of  a 

single  variable 146 

CHAPTER   X. 
Integration. 


94.  Integration  and  integral  defined.     Notation   . 

95.  Examples  of  the  summation  of  infinitesimals 

96.  Integration  as  summation.     The  definite  integral  . 

97-   Integration  as  the  inverse  of  differentiation.  The  indefinite  in 
Constant  of  integration.     Particular  integrals 

98.  Geometric  or  graphical  representation  of  definite  integrals 
Properties  of  definite  integrals 

99.  Geometric  or  graphical  representation  of  indefinite  integrals 
Geometric  meaning  of  the  constant  of  integration  . 

100.  Integral  curves 

101.  Summary 


tegi-al 


148 
150 
154 
160 
162 
163 
164 
166 
167 
168 
169 


CONTENTS.  XUl 


CHAPTER   XI. 


Elementary  Integrals. 

ART.  PAOK 

103.  Elementary  integrals 172 

104.  General  theorems  in  integration 173 

105.  Integration  aided  by  substitution 175 

106.  Integration  by  parts        . 177 

107.  Further  elementary  integrals 180 

108.  Integration  of  f{x)dx  when  f{x)  is  a  rational  fraction  .                 .  184 

109.  Integration  of  a  total  differential 188 


CHAPTER   XII. 

Simple  Geo3ietrical  Applications  of  Integration. 

111.  Areas  of  curves :  Cartesian  coordinates 192 

112.  Volumes  of  solids  of  revolution 199 

113.  Derivation  of  the  equations  of  curves 203 

CHAPTER   XIII. 
Integration  of  Irrational  and  Trigonometric  Functions. 

Integration  of  Irrational  Functions. 

115.  The  reciprocal  substitution 206 

116.  Differential  expressions  involving  y/a  +  hx 207 

117.  A.   Expressions  of  form  F{x^  Vx^  +  ax  +  b)dx.     B.   Expressions 

of  form  F(x,  V-x2  +  ax  +  h)dx 208 

118.  Tofind  fx"»(a  +  6x")Pda: 211 

Integration  of  Trigdnometric  Functions. 

119.  Algebraic  transformations 215 

120.  Integrals  reducible  to  |  F(ii)du,  in  which  u  is  one  of  the  trigo- 

nometric ratios 216 

121.  Integration  aided  by  multiple  angles 217 

122.  Reduction  formulas 218 


xiv  CONTENTS. 


CHAPTER   XIV. 


Approximate  Integration.     Mechanical  Integration. 

ART.  PAGE 

123.  Approximate  integration  of  definite  integrals          ....  223 

124.  Trapezoidal  rule  for  measuring  areas  and  evaluating  definite  inte- 

grals      223 

125.  Parabolic  rule  for  measuring  areas  and  evaluating  definite  integrals  225 

126.  Integration  in  series 227 

127.  Mechanical  devices  for  integration 228 


CHAPTER  XV. 

Successive  Integration.     Multiple  Integrals.  •  Applications. 

129.    Successive  integration :  one  variable.     Applications       .         .         .  230 

180.    Successive  integration  :  several  variables 232 

131.  Finding  areas :  rectangular  coordinates 234 

132.  Finding  volumes :  rectangular  coordinates 235 

133.  Finding  volumes :  polar  coordinates 238 


CHAPTER  XVI. 

Further  Geometrical  Applications  of  Integration. 

135.  Volumes  of  solids  of  known  cross-section 240 

136.  Areas :  polar  coordinates 24? 

137.  Lengths  of  curves  :  rectangular  coordinates 245 

138.  Lengths  of  curves :  polar  coordinates     ......  248 

139.  Areas  of  surfaces  of  revolution 249 

140.  Areas  of  surfaces  z  =  /(ic,  y) 253 

141.  Mean  values 255 

CHAPTER   XVII. 

CoNCAViTr  and  Convexity.    Contact  and  Curvature.    Evolutes 
AND  Involutes. 

142.  Concavity  and  convexity  :  rectangular  coordinates         .        .        .  260 

143.  Order  of  contact 261 

144.  Osculating  circle 264 

145.  The  notion  of  curvature 265 

146.  Total  cuiTature.    Average  curvature.    Curvature  at  a  point        .  266 


CONTENTS.  XV 

ART.  PAOB 

147.  The  curvature  of  a  circle 267 

148.  To  find  the  curvature  at  any  point  of  a  curve  :  rectangular  coordi- 

nates      267 

149.  The  circle  of  curvature  at  any  point  of  a  curve       ....  268 

150.  The  radius  of  curvature :  polar  coordinates    .         .         •        .         .271 

151.  Evolute  of  a  curve 272 

152.  Properties  of  the  evolute          .        .         .         .         .         .         .         .  273 

153.  Involutes  of  a  curve 276 

CHAPTER   XVIII. 
Special  Topics  relatixg  to  Curves. 

154.  Family  of  curves.     Envelope  of  a  family  of  curves         .         .         .  277 

155.  Locus  of  ultimate  intersections  of  the  curves  of  a  family        .         .  278 

156.  Theorem 280 

157.  To  find  the  envelope  of  a  family  of  curves  having  one  parameter  .  281 

158.  Envelope  of  a  family  of  curves  having  two  parameters  .         .         .  284 

159.  Rectilinear  asymptotes 286 

160.  Asymptotes  parallel  to  the  axes 288 

161.  Oblique  asymptotes 290 

162.  Rectilinear  asymptotes :  polar  coordinates 292 

163.  Singular  points 293 

164.  Multiple  points 293 

165.  To  find  multiple  points,  cusps,  and  isolated  points         .        .         .  296 

166.  Curve  tracing 298 

CHAPTER   XIX. 
Infinite  Series. 

167.  Infinite  series :  definitions,  notation 300 

168.  Questions  concerning  infinite  series 301 

169.  Study  of  infinite  series 303 

170.  Definitions.     Algebraic  properties  of  infinite  series        .         .         .  304 

171.  Tests  for  convergence 307 

172.  Integration  of  infinite  series 310 

173.  Differentiation  of  infinite  series 312 

174.  Applications  of  the  integration  and  differentiation  of  series  .        .  313 

CHAPTER  XX. 
Taylor's  Theorem. 

176.  Derivation  of  Taylor's  theorem 318 

177.  Another  form  of  Taylor's  theorem 323 


XVI  CONTENTS. 

AKT.  PAGK 

178.  Maclaurin's  theorem  and  series 324 

179.  Relations  between  the  circular  functions  and  exponential  functions  327 

180.  Another  method  of  deriving  Taylor's  and  Maclaurin's  series  .  329 

181.  Application  of  Taylor's  theorem  to  the  determination  of  condi- 

tions for  maxima  and  minima 331 

182.  Application  of  Taylor's  theorem  to  the  deduction  of  a  theorem  on 

the  contact  of  curves 332 

183.  Applications  of  Taylor's  theorem  in  elementary  algebra        .         .     333 

CHAPTER   XXI. 
Differential  Equations. 

184.  Definitions.     Classifications.     Solutions 384 

185.  Constants  of  integration.     General  solutions.     Particular  solutions    335 

Equations  of  the  First  Order. 

186.  Equations  of  the  form /(a;) (Zx  +  i^(2/)^?/ =  0 335 

187.  Homogeneous  equations 336 

188.  Exact  differential  equations.     Integrating  factors  .        .         .  336 

189.  The  linear  equation 337 

190.  Equations  not  of  the  first  degree  in  the  derivative : 

The  form  x  =  f(y,  p)  ;  the  form  y  =  /(x,  p)  ;  Clairaut's  equation    338 

191.  Singular  solutions ."        .         .     340 

192.  Orthogonal  trajectories .         .     341 

Equations  of  the  Second  and  Higher  Orders. 

193.  Linear  equations  with  constant  coefficients.     Homogeneous  linear 

equations 346 

194.  Special  equations  of  the  second  order  : 


;^^=-^w 


>/(S-l-)=«-<S'i-)-     •    •- 


APPENDIX. 

Note  A.    Hyberbolic  functions .         .  353 

Note  B.    Intrinsic  equations 363 

Note  C.    Indeterminate  forms 367 

Note  D.    Applications  to  mechanics 372 

Collection  of  Examples ,        .        .        .  381 


CONTENTS.  XVll 

PAGE 

Integrals  for  Review  Exercises  and  for  Reference    .         .         .  401 

Figures 409 

Answers 415 

Index 433 

SHORT   COURSE 

FOR  STUDENTS   HAVING  A   MINIMUM  OF  TIME 

(The  Roman  numerals  refer  to  chapters,  the  Arabic  to  articles.) 

I.  ;  11.  ;  III.  17, 18,  21-27  a;  IV.  ;  V.  57-65 ;  VI.  68-70  ;  VII.  ;  VIII.  79-84, 
86  ;  (IX.)  ;  X.  ;  XI.  ;  XII.  ;  (XIII.  114-116,  118-122)  ;  XIV.  ;  XV.  ; 
XVI.  1.34-139,  141  ;  XIX.  (167-171),  174  ;  XX.  175,  180,  Exs.  176- 
178  ;  XVII.  ;  XVIII. 


o 


INFINITESIMAL   CALCULUS. 

CHAPTER   I. 

INTRODUCTORY   PROBLEMS. 

1.  The  infinitesimal  calculus  is  one  of  the  most  powerful  mathe- 
matical instruments  ever  invented.*  Many  practical  problems  can 
be  solved  by  its  means  with  wonderful  ease  and  rapidity.  Even 
a  slight  acquaintance  with  the  calculus  is  very  helpful  in  the  study 
of  many  other  subjects,  for  example,  geometry,  astronomy,  physics, 
and  engineering;  and  the  fullest  knowledge  possible  about  the 
calculus  is  necessary  for  advance  in  these  subjects.  Some  of 
the  higher  branches  of  mathematics  consist  largely  of  special 
investigations  in  the  infinitesimal  calculus  and  extensions  of  its 
principles,  methods,  and  applications.! 

In  this  book  the  fundamental  notions  and  principles  of  the 
calculus  are,  to  a  certain  extent,  explained,  and  applications  are 
made  to  the  solution  of  some  simple  practical  problems.  As  a 
preliminary  to  the  study  there  is  in  this  chapter  a  discussion  of 
a  few  problems.  This  discussion  introduces  in  an  informal  way 
the  notions  and  principles  and  methods  which  are  at  the  founda- 
tion of  the  infinitesimal  calculus,  and  also  provides  material  which 
serves  to  illustrate  a  few  of  the  articles  that  follow. J 

*  The  calculus  as  used  to-day  was  invented  independently  by  Newton  and 
Leibnitz.     See  Art.  94,  note. 

t  The  word  "  infinitesimal  "  serves  to  distinguish  the  subject  from  other 
branches  of  mathematics,  such  as  the  calculus  of  finite  differences,  the  calculus 
of  variations,  the  calculus  of  quaternions,  etc. 

t  An  important  fact  in  the  history  of  the  calculus  is  that  the  problems  in 
Arts.  3-6  were  the  occasion  of  the  invention  and  development  of  some  divisions 
of  the  subject. 

1 


2  INFINITESIMAL   CALCULUS,  [Ch.  I. 

Note.  A  knowledge  of  the  meaning  of  the  term  speed  or  rate  of  motion  is 
presupposed  in  the  following  two  articles.  If  a  body  moves  tlirough  equal 
distances  in  equal  times,  it  is  said  to  have  uniform  speed.  The  average  speed 
of  a  body  during  the  time  that  it  is  moving  through  a  certain  distance,  is  the 
uniform  speed  at  which  a  body  will  pass  over  that  distance  in  that  time. 
For  instance,  if  a  bicyclist  wheels  30  miles  in  3  hours,  his  average  speed  is 
12  miles  per  hour  ;  if  a  body  moves  through  45  feet  in  5  minutes,  its  average 
speed  is  9  feet  per  minute.  The  number  which  indicates  the  average  speed 
of  a  body  while  it  is  moving  through  a  certain  distance,  is  the  ratio  of  "the 
number  of  units  of  length  in  the  distance  to  the  number  of  units  of  time  spent 
during  the  motion.  In  other  words,  the  measure  of  the  speed  is  the  ratio  of 
the  measure  of  the  distance  to  the  measure  of  the  corresponding  time.  Thus, 
in  the  instances  above,  12  =  36  :  3,  9  =  45  :  5. 

Any  reader  of  this  book  knows  what  is  meant  by  the  statements  that  a 
train  is  running  at  a  particular  instant  at  the  rate  of  30  miles  an  hour,  and 
that  at  another  instant,  some  minutes  later  say,  it  is  running  at  the  rate  of  40 
miles  an  hour.  This  notion,  viz.  the  speed  of  a  moving  body  at  a  par- 
ticular instant,  will  be  developed  further  by  the  examples  that  follow. 

2.  Speed  of  a  moving  train.  Suppose  that  a  person  is  standing 
by  a  railway  and  wishes  to  ascertain  the  speed  at  which  a  train 
is  going  by  him.  A  way  to  determine  this  speed  approximately 
Avould  be  to  find  the  distance  passed  over  in  five  seconds  by  the 
train,  or  by  a  definite  mark  on  the  train,  say  a  vertical  line.  (The 
place  where  the  observer  stands  may  be  at  one  end  of,  or  upon, 
the  measured  distance.)  If  the  observer  knew  the  distance  passed 
over  in  three  seconds,  he  would  get  the  speed  .more  accurately ; 
yet  more  accurately,  if  he  knew  the  distance  passed  over  in  one 
second;  more  accurately  still,  if  he  knew  the  distance  passed 
over  in  half  a  second;  and  so  on.  The  point  to  be  noted  and 
emphasised  in  this  illustration  is  this :  the  less  the  time  and  the 
corresponding  distance  that  can  be  observed,  the  more  nearly  will 
the  observer  obtain  the  actual  speed  of  the  train  just  at  the 
moment  when  it  is  passing  him. 

3.  To  determine  the  speed  of  a  falling  body.  Let  a  body  fall 
vertically  from  rest.  It  is  known  that  in  t  seconds  from  the 
time  of  starting,  the  body  passes  through  ^gf^  feet.  (Here  g 
denotes  a  number  whose  approximate  value  is  32.2.)  That  is,  if 
s  denotes  the  number  of  feet  through  which  the  body  falls  in  t 
seconds,  s  =  i  at'K 


2,  3.] 


INTRODUCTORY  PROBLEMS. 


As  the  body  descends  its  speed  is  continually  changing  and  grow- 
ing greater;  but  at  any  particular  instant  it  has  some  definite 
speed.  Let  it  be  required  to  find  the  speed  after  it  has  been 
falling  for :  (a)  4  seconds ;  (b)  t^  seconds. 

(a)  To  find  speed  after  the  bodt/  has  been  falling  from  rest  for  4  seconds. ' 
A  method  of  getting  an  approximate  value  of  this  speed  is  as  follows.  Find 
the  distance  through  which  the  body  would  fall  in  4  seconds ;  then  find  the 
distance  through  which  it  would  fall  in  a  little  more  than  4  seconds.  There- 
from deduce  the  average  value  of  the  speed  from  the  end  of  the  fourth  second 
to  the  last  instant  (Note,  Art.  1).  This  average  speed  may  be  taken  as  an 
approximate  value  of  the  speed  at  the  end  of  the  fourth  second.  The  smaller 
the  interval  of  time  which  is  taken  after  the  fourth  second,  the  more  nearly 
will  the  average  speed  for  the  interval  be  equal  to  the  actual  speed  just  at 
the  end  of  the  fourth  second.  This  is  also  apparent  from  the  following 
calculations : 


1  . 

a  o 

fi 

3 

Length  of  fall, 
in  feet. 

Increase  in   time 
after  4  seconds. 
(In  seconds.) 

C'orrespoiidin'r 

increa.'it-  in 

distance, 

in  feet. 

Average  speed  during  increased 
time,  in  feet  per  second. 

4. 

8r/ 









4.1 

8.405  <7 

.1 

.405  sr 

4.05  gr       or 

130.41 

4.01 

8.04005^ 

.01 

.04005  </ 

4.005(7 

128.961 

4.001 

8.0040005^ 

.001 

.0040005  c/ 

4.0005(7 

128.8161 

4.0001 

8.000400005^ 

.0001 

.000400005gr 

4.00005  g 

128.80161 

i  +  h 

(8  +  4/i  +  |/i2)^ 

h 

{^h^\h^)g 

(-i> 

128.8  + 16.1  xfe 

It  is  evident  that  the  less  the  increase  given  to  the  4  seconds,  the  more 
nearly  does  the  average  speed  during  this  additional  time  approach  to  128.8 
feet  per  second.  The  last  line  of  the  table  shows  that,  no  matter  how  short 
a  time  h  may  be,  the  average  speed  during  this  time  has  a  definite  value, 
namely  (128.8  +  16.1  x  h)  feet  per  second.  The  number  in  brackets  becomes 
more  and  more  nearly  equal  to  128.8  when  h  is  made  smaller  and  smaller ;  the 
difference  between  it  and  128.8  can  be  made  as  small  as  one  pleases,  merely 
by  decreasing  7i,  and  will  become  still  less  when  h  is  further  diminished. 
Since  the  number  (128.8  +  16.1  x  h)  behaves  in  this  way,  the  speed  of  the 
falling  body  at  the  end  of  the  fourth  second  is  manifestly  128.8  feet  per 
second. 


4  INFINITESIMAL   CALCULUS.  [Ch.  I. 

(6)  To  find  the  speed  after  the  body  has  been  falling  for  ti  seconds.  Let 
si  denote  the  distance  in  feet  through  which  the  body  has  fallen  in  the  ti 
seconds.     It  is  known  that  ^   _  i  ^^  2  (i\ 

Let  A^i  (read  "  delta  ii ")  denote  any  increment  given  to  tu  and  Asi  denote 
the  corresponding  increment  of  si. 

Note  1.  Here  A^i  does  not  mean  A  x  ^1.  The  symbol  A  is  used  with  a 
quantity  to  denote  any  difference,  change,  or  increment,  positive  or  negative 
(i.e.  any  increase  or  decrease),  in  the  quantity.  Thus  Ax  and  Ay  denote 
"  increment  of  x,""  "  increment  of  y,"  "  difference  in  x,"  "  difference  in  y." 

Then  si  +  Asi  =  i  g(ti  +  A^i)'^.  (2) 

Hence,  by  (1)  and  (2),  Asi  =  gti  •  A«i  +  ^  9'(A«i)2. 

.'.^=gt,-]-ig-Ati.  (3) 

Ati 

Here  =^  is  the  average  speed  for  the  time  Ati  and  the  corresponding 

A«i  Asi 

distance   Asi.      Now  the  sihaller  Ati    is   taken,  the   more  nearly  will  ^ 

approximate  to  the  actual  speed  which  the  falling  body  has  at  the  end  of 
the  ^ith  second.  But  when  A^i  is  taken  smaller  and  smaller  (in  other  words, 
when  Ati  approaches  nearer  and  nearer  to  zero),  the  second  member  of  equa- 
tion (3)  approaches  nearer  and  nearer  to  gti.     Equation  (3)  also  shows  that 

=^  can  be  made  to  differ  as  little  as  one  pleases  from  gti,  merely  by  taking 

Ati 

Ati  small  enough.     Hence  it  is  reasonable  to  conclude  that  at  the  end  of  the 

^ith  second 

the  speed  of  the  falling  body  =  gti  feet  per  second.  (4) 

Here  ti  may  be  any  value  of  t.  So  it  is  usual  to  express  conclusion  (4) 
thus :  the  speed  of  a  body  that  has  been  falling  for  t  seconds  is  gt  feet  per 
second.  This  result  (speed  =  gt  feet  per  second)  is  a  general  one,  and  can 
be  applied  to  special  cases.  Thus  at  the  end  of  the  fourth  second  the  speed 
is  g^  X  4  or  128.8  feet  per  second,  as  found  in  (a)  ;  at  the  end  of  10  seconds 
the  speed  is  10  g  or  322  feet  per  second. 

The  two  principal  points  to  be  noted  in  this  illustration  are : 

(1)  No  matter  what  the  value  of  A^i  may  be,  or  how  small  A^i 
may  be,  the  quantity  — ^  has  a  definite  value,  namely,  gti  +  \g  -  A^j ; 

(2)  When  A^i  is  taken  smaller  and  smaller,  — ^  gets  nearer  and 

nearer  to  gt^ ;  and  the  difference  between  them  can  be  made  as 
small  as  one  pleases  by  giving  Afi  a  definite  small  value;  this 
difference  remains  less  than  the  assigned  value  when  A^j  further 
decreases. 


4.]  INTBODUCTORT  PROBLEMS.  6 

Note  2,  The  definite  small  value  referred  to  in  (2)  can  be  easily  found. 
For  example,  suppose  that  — ^  is  to  differ  from  gt\  by  not  more  than  k  say  {k 
being  any  small  quantity,  as  a  millionth,  or  a  million-millionth). 

Then  ^'  -  gh  ^  k.      But  ~-9h  =  ig'  Ah  by  (3).        ^ 

—  —2k 

.  •.   ^-  g  •  Ati^k;  accordingly  A^i  <  —  • 

Note  3.     It  should  be  observed,  as  shown  by  equation  (3),  that  the  value  of 

-^  depends  upon  the  values  of  both  ti  and  A^i.     On  the  other  hand,  the 

Ah  Asi 

value  to  which  —  tends  to  become  equal  as  Ah  decreases,  depends  (see  (4)) 

upon  h  alone.     The  quantity  A^i  is  any  increment  whatever  of  h,  but  it  does 
not  depend  upon  the  value  of  h- 

4.  To  determine  the  slope  of  the  tangent  to  the  parabola  y  =  x-  : 
(a)  at  the  point  whose  abscissa  is  2 ;  (6)  at  the  point  whose  abscissa 
is  x^.  v\ 

(a)  Let  VOQ,  Fig.  1,  be  the 
parabola  y  =  x^,  and  P  be  the 
point  whose  abscissa  is  2. 
Draw  the  secant  PQ.  If  PQ 
turns  about  P  until  Q  coin- 
cides with  P,  then  PQ  will 
take  the  position  PT  and  be- 
come the  tangent  at  P.  The 
angle  QPE  will  then  become  the  angle  TPB. 

Note  1.  This  conception  of  a  tangent  to  a  curve  has  probably  been 
already  employed  by  the  student  in  finding  the  equations  of  tangents  to  circles, 
parabolas,  ellipses,  and  hyperbolas.  The  process  generally  followed  in  the 
analytic  treatment  of  the  conic  sections  is  as  follows  :  The  equation  of  the 
secant  PQ  is  found  subject  to  the  condition  that  P  and  Q  are  on  the  curve  ; 
then  Q  is  supposed  to  move  alo7ig  the  curve  until  it  reaches  P.  The  resulting 
form  of  the  equation  of  the  secant  is  the  equation  of  the  tangent  at  P.  The 
calculus  method  (now  to  be  shown)  of  finding  tangents  to  curves  is  preferred 
by  some  teachers  of  analytic  geometry;  e.g.  see  A.  L.  Candy,  Analytic 
Geometry,  Chap.  V. 

Draw  the  ordinates  PL  and  QM-,  draw  PM  parallel  to  OX. 
Let  PR  be  denoted  by  Ace,  and  EQ  hy  Ay.     Then  the  slope  of 

the  secant  FQ  is  ^-    f  For  tan  EPQ  =  ^-^ 
Ax      \  PE  J 


Q. 

^ 

/^ 

y     AX 

R 

/G  L       M 
Fig.  1. 


INFINITESIMAL   CALCULUS. 


[Ch.  L 


The  following  table  shows  the  value  of  -^  for  various  values 


of  Aic. 


Ax 


Value  of  X. 

Corresponding 
value  of  y. 

(Increase  over  a;). 

Ay 

(Increase  over  y). 

Corresponding 

value  of  ^y. 
Xv. 

2. 

4. 

_ 

2.1 

4.41 

.1 

.41 

4.1 

2.01 

4.0401 

.01 

.0401 

4.01 

2.001 

4.004001 

.001 

.004001 

4.001 

2.0001 

4.00040001 

.0001 

.00040001 

4.0001 

2-\-h 

A  +  ^h  +  h^ 

h 

4  /i  +  /i2 

4  +  /^ 

It  is  apparent  from  this  table  that  the  less  Ax  is,  the  more  nearly  does 

—2  approach  the  value  4.    The  last  line  shows  that,  no  matter  how  small  Ax 

Ax  ^ 

(or  K)  may  be,  -^  has  a  definite  value,  namely  A-\-h.    This  number  becomes 

Ax 
more  and  more  nearly  equal  to  4  when  h  is  made  less  and  less  ;  the  difference 
between  it  and  4  can  be  made  as  small  as  one  pleases,  merely  by  decreasing  h 
to  a  certain  definite  value,  and  will  continue  to  be  as  small  or  smaller  when 
h  is  further  diminished.     Because  the  number  \  -\-  h  behaves  in  this  way, 

it  is  evident  that  — ^  will  reach  the  value  4  when  Ax  decreases  to  zero. 

Ax 
Accordingly  the  slope  of  the  tangent  PT  is  4  ;  and  hence  angle   TPR  or 
PWL  is  75°  57'  49". 

(h)  To  determine  the  slope  of  the  tangent  at  the  point  whose 
abscissa  is  x^. 

Let  (Fig.  1)  P  be  the  point  (xi,  yi).  Draw  the  secant  PQ,  and  the 
ordinates  PL  and  Q3I ;  draw  PB  parallel  to  OX.  Let  PB,  the  difference 
between  the  abscissas  of  P  and  Q,  be  denoted  by  Axi,  and  let  BQ,  the 
difference  between  the  ordinates  of  P  and  Q,  be  denoted  by  Ayi.     Then 

tangent  QPB=^  =  ^^-1. 
^  PB     Axi 

If  Q  be  moved  along  the  curve  toward  P,  the  secant  PQ  will  approach 
the  position  of  PT,  the  tangent  at  P ;  at  last,  when  Q  reaches  P,  the  secant 
PQ  becomes  the  tangent  PT.  As  Q  approaches  P,  Axi  becomes  less  and 
less,  and  when  Q  reaches  P,  Axi  becomes  zero.  Conversely,  as  Axi  decreases, 
PQ  approaches  the  position  PT.  Accordingly,  the  slope  of  the  tangent  PT 
can  be  determined  by  finding  what  the  slope  of  the  secant  PQ,  namely  -^- 
approaches  when  Axi  approaches  zero. 


Axi 


4.]  INTRODUCTORY  PROBLEMS,  7 

yi  +  Ayi{=  MQ)  =  (xi -^  Axiy. 
Hence,  on  subtraction,  Ayi  =  2  xi  •  Axi  +  (Axi)2.  (1) 

...  :^  =  2  a:i  +  Axi.  (2) 

Axi 

This  equation  shows  that  -^  approaches  nearer  to  2  xi  when  Axi  decreases. 

It  also  shows  that  ^^  can  be  made  to  differ  as  little  as  one  pleases  from  2  xi, 

Axi 
merely  by  taking  Axi  small  enough,  and  that  this  difference  will   become 
smaller  when  Axi  is  further  diminished.     (For  instance,  if  it  is  desired  that 

^  —  2  xi  be  less  than  any  positive  small  quantity,  say  e,  it  is  only  necessary 

Axi 

to  take  Axi  less  than  e.)     Accordingly. 

the  slope  of  PT  (the  tangent  at  P)  =  2  xi.  (3) 

The  two  principal  points  to  be  noted  in  tliis  illustration  are : 

(1)  No  matter  what  the  value  of  Ax^  may  be,  or  how  small  Aa?i 

may  be,  the  quantity  -^*  has  a  definite  value,  namely  2  x^  H-  A4. 

^^  A?/ 

(2)  When   Ax^   decreases,   the    quantity   -—  approaches    the 

Av,  ^1 

value  2  x^ :  the  difference  between  — ^  and  2  Xi  can  be  made  as 

^ '.  AXi 

small   as  any  number  that  may  be  assigned,  by  giving  Ax-^  a 

definite    small    value ;    this    difference    remains    less    than    the 

assigned  value  when  Ax^  further  decreases. 

Avi 
Note  1.     The  value  of  -^,  as  shown  by  Equation  (2),  depends  upon  the 

A 

values  of  both  xi  and  Axi.    On  the  other  hand,  the  value  to  which  -~^ 

tends  to  become  equal  as  Axi  decreases,  depends  (Equation  (3))  upon  xi 
alone.  The  value  of  Axi  does  not  depend  upon  the  value  of  Xi ;  for  Q 
(Fig.  1)  may  be  taken  anywhere  on  the  curve. 

Note  2.  The  method  used  in  getting  result  (3)  does  not  depend  upon 
the  particular  value  of  Xi.  The  result  is  perfectly  general,  and  may  be 
expressed  thus  :  '''■the  slope  of  the  curve  y  =  x?  is  2  x."  This  general  result 
can  be  used  for  finding  the  slope  at  particular  points  on  the  curve.  For 
instance,  if  Xi  =  2,  the  slope  is  4,  as  found  in  {a) ;  if  Xi  =—  1,  the  slope 
is  —  2,  and  accordingly,  the  angle  made  by  the  tangent  with  the  x-axis  is 
116°  34'.     (It  is  advisable  to  make  a  figure  showing  this.) 

Note  3.  In  the  infinitesimal  calculus,  as  well  as  in  other  branches  of 
mathematics,  it  is  very  important  for  the  student  always  to  have  a  clear 


8  INFINITESIMAL   CALCULUS.  [Ch.  I. 

understanding  of  the  meaning  of  the  operations  which  he  performs  with 
numbers^  and  to  interpret  rightly  the  numerical  results  obtained  by  these  oper- 
ations. Thus,  if  it  is  stated  tliat  6  men  work  5  days  at  2  dollars  per  day  each, 
the  numbers  6,  5,  and  2  are  treated  by  the  operation  called  multiplication, 
and  the  number  60  is  obtained.  The  calculator  then  applies,  or  interprets, 
this  numerical  result  as  meaning,  not  60  men,  or  60  days,  but  that  the  men 
have  earned  60  dollars.  In  the  curve  above,  y  —  y?.  This  does  not  mean 
that  at  any  point  on  the  curve  the  ordinate  is  equal  to  the  square  on  the 
abscissa,  i.e.  a  length  is  equal  to  an  area.  By  y  =  x^  it  is  meant  that  the 
number  of  units  of  length  in  any  ordinate  is  equal  to  the  square  of  the  num- 
ber of  units  of  length  in  the  corresponding  abscissa.  Again,  the  result  in 
Equation  (3)  does  not  mean  that  the  slope  of  FT  is  twice  OL.  The  result 
means  that  the  number  which  is  the  value  of  the  trigonometric  tangent  of 
the  angle  TFB  is  twice  the  number  of  units  of  length  in  OL. 

Many  persons  who  can  perform  operations  of  the  calculus  easily  and 
accurately,  cannot  correctly  or  confidently  interpret  the  results  of  these 
operations  in  concrete  practical  problems  in  geometry,  physics,  and  engi- 
neering. Thus,  some  engineers  who  have  had  a  fairly  extended  course  in 
calculus  discard  it  when  possible,  and  solve  practical  problems  by  much 
longer  and  more  laborious  methods.  Such  a  misfortune  will  not  happen  to 
those  who  early  get  into  the  habit  of  giving  careful  thought  to  finding  out  the 
real  meaning  of  the  operations  and  results  of  the  calculus.  They  will  not 
only  "understand  the  theory,"  but  they  can  use  the  calculus  as  a  tool  with 
ease  and  skill. 

Note  4.  In  Fig.  1  let  a  point  Qi  be  taken  on  the  curve  to  the  left  of  P, 
and  draw  the  secant  Q\F.  (The  drawing  for  this  note  is  left  to  the  student.) 
It  is  obvious  from  the  figure  that  the  same  tangent  PT  is  obtained,  whether 
the  secant  QiP  revolves  until  Q-^  reaches  P,  or  QP  revolves  until  Q  reaches 
P.  This  may  also  be  deduced  algebraically.  Let  the  coordinates  of  ^i  be 
Xi  —  Acci,  y\  —  A^i.  [Here  the  ^Xx  and  Ayi  are  not  necessarily  the  same  in 
amount  as  the  Axi  and  A«/i  in  (6).]     Draw  the  ordinate  QiM\.    Then 

yx{=LP)=x^% 

yi  -  Ayi  (=  MiQi)  =  (xi  -  Axi)2. 

Whence,  it  follows  that  — ^  =  2  cci  —  Axi. 

AXi 

Accordingly,  when  AiCi  approaches  zero,  -^  approaches  the  value  2xi. 

Axi 

Note  5.  Thoughtful  beginners  in  calculus  are  frequently,  and  not  un- 
naturally, troubled  by  the  consideration  that  when  A^i  (Art.  3  b)  is  diminished 

to  zero,  ~  has  the  form  -  ;    and  likewise,  when  Axi  (Art.  4  b)  becomes 
A^i  0 

zero,  -^  becomes  ^.    It  is  true  that  ^  is  indeterminate  in  form :  and,  if 
Axj  V  0 


4.]  INTRODUCTORY  PROBLEMS.  9 

it  is  presented  without  ariy  information  being  given  concerning  the  whence 
and  the  wherefore  of  its  appearance,  then  its  value  cannot  be  determined. 
In  the  cases  in  Arts.  3,  4,  however,  there  is  given  information  which  makes 

it  possible  to  tell  the  meaning  of  the  quantity  ^  that  appears  at  the  final  stage 

of  each  of  these  problems.     In  these  cases  one  knows  how  the  quantities 

-—  and  -r^  are  behaving  when  Mi  and  Axi  respectively  are  approaching 

zero ;  and  by  means  of  this  knowledge  he  can  confidently  and  accurately 
state  what  these  ratios  will  become  when  A^i  and  Aa^i  actually  reach  zero.* 

Note  6.     Moreover,  it  should  be  carefully  noted  that  at  the  final  stages 

in  the  solution  of  the  problems  in  Arts.  3  and  4,  ~  is  not  regarded  as  a 

A  61 

fraction  composed  of  tico  quantities,  Asi  and  A«i,  but  as  a  single  quantity, 

namely  the  speed  after  ti  seconds ;  likewise,  that  ~~  is  then  not  regarded 

as  a  fraction  at  all,  but  as  a  single  quantity,  namely  the  slope  of  the  tangent 
at  P. 

Note  7.  The  student  should  not  be  satisfied  until  he  clearly  perceives, 
and  understands,  that  the  method  employed  in  solving  the  problems  in 
Arts.  3  and  4  is  not  a  tentative  one,  but  is  general  and  sure,  and  that  the 
results  obtained  are  not  indefinite  or  approximate,  but  are  certain  and  exact. 


EXAMPLES. 

1.  Assuming  the  result  in  Art.  4  (6),  namely,  that  the  slope  of  the  tangent 
at  a  point  (xi,  y{)  on  the  curve  y  =  x^  is  2a:i,  find  the  slope  and  the  angle 
made  with  the  ic-axis  by  the  tangent  at  each  of  the  points  whose  abscissas  are 

.5,   0,    1,    1.5,   2,   2.5,    3,   4,    -2,    -3,    -  f,    - 1,    -  f . 

2.  In  the  curve  in  Ex.  1  find  the  coordinates  of  the  points  the  tangents  at 
which  make  angles  of  20°,  30°,  45°,  60°,  85°,  115°,  145°,  160°,  170°,  respec- 
tively, with  the  a;-axis. 

3.  Draw  figures  of  the  following  curves.     Find  the  value  of  —  at  any 

Ay 
point  (x,  y)  in  the  case  of  each  curve  ;   then  find  what  --  is  approaching 

when  Ax  approaches  zero : 

(a)   x2  +  2/2  =  16;        (b)  y  =  x^ -[- x -\- 1;  (c)  y  =  x^  ; 

(d)   y^  =  Sxi                (e)  9x2  +  16y2  =  144  ;  (/)  9 .x2  _  16 1/2  =  144  ; 

(9)   y^  =  ^px]              (h)  b'^x'^  +  a:^y^  =  a'^b'^ ;  (i)  b^x^  -  a^y'^  =  a'^b^. 

*  The  mathematical  phraseology  and  notation  employed  to  express  these 
ideas  is  given  in  Chapter  II. 


10 


IN  FINITE  SIM  A  L   CALC  UL  US. 


[Ch.  I. 


TSuGGESTiON.     In  (a),  (x  +  Ax)2  +  (2/  +  A2/)2  =  16.     It  can  then  be  de- 

*-  Ay         2x4- Ax  1 

ducedthat^  =  -2^^^.J 

Compare  the  results  found  in  (g),  (h),  and  (i),  with  those  found  in 
analytic  geometry. 

4.  Using  the  results  obtained  in  Ex.  3,  find  the  slopes  and  the  angles  made 
with  the  X-axis  by  the  tangents  in  the  following  cases  : 

(a)    The  curve  in  Ex.  3  (a),  at  the  points  whose  abscissas  are 

4,   2,    1,    0,   -  1.5,   -  3.5. 

(&)   The  curve  in  Ex.  3  (c),  at  the  points  whose  abscissas  are 

-3,   -2,-1,    0,    1.5,  2.5. 

(c)  The  curve  in  Ex.  3  (d),  at  the  points  whose  abscissas  are 

0,    1,   2,   3,    6,    8. 

(d)  The  curve  in  Ex.  3  (e),  at  the  points  whose  abscissas  are 

0,    1,    2,   4,   -  .5,    -  1.5. 

(e)  The  curve  in  Ex;.  3  (/),  at  the  points  whose  abscissas  are 

4,   8,    10,    -5,    -7. 

5.  Using  the  results  obtained  in  Ex.  3,  find  the  points  on  the  curve  in 
Ex.  3  (a)  the  tangents  at  which  make  angles  40"^  and  136°  with  the  x-axis. 

6.  Do  as  in  Ex.  5  for  the  curves  whose  equations  are  given  in  Ex.  3  (c), 
(d),  (e),  and  (/). 

7.  Do  some  of  the  examples  in  Art.  59.     Make  careful  drawings  in  each 


5.   To  determine  the  area  of  a  plane  figure.     A  plane  area,  say 
ABCD,  may  be  supposed  to  be  divided  into  an  infinitely  great 

number  of  infinitely  small  rec- 
tangles. It  will  be  seen  later 
that  the  limit  of  the  sum  of  these 
rectangles  when  they  are  taken 
smaller  and  smaller,  is  the  area. 
The  calculus  furnishes  a  way  to 
find  this  limit.  Even  at  this 
stage  in  the  study  of  the  calculus 
the  student  can  get  some  useful 
ideas  concerning  this  problem  by  making  a  brief  inspection  of 
Art.  95,  Exs.  (a),  (h),  (c).     [Art.  14  discusses  the  term  "limit."] 


5-7.]  INTBODUCTOBY  PBOBLEMS,  11 

6.  (a)  To  find  a  function  when  its  rate  of  change  at  any  (every) 
moment  is  known,  or,  in  more  general  terms,  when  its  laic  of  change 
is  known.  In  Art.  3  (5)  a  particular  example  has  been  given  of 
this  general  problem,  viz.  to  determine  the  rate  of  change  of  a  func- 
tion at  any  moment.  The  calculus  not  only  provides  a  method  of 
solving  this  general  problem,  but  also  provides  a  method  of  solving 
the  inverse  problem  which  is  stated  above. 

(6)  To  find  the  equation  of  a  curve  when  its  slope  at  any  (every) 
point  is  known.  In  Art.  4  (6)  a  particular  example  has  been  given 
of  this  general  problem,  viz.  to  determine  the  slope  of  a  curve  at 
any  point  on  it.  The  calculus  not  only  provides  a  method  of 
solving  this  problem,  but  it  also  provides  a  method  of  solving  the 
inverse  problem  which  has  just  been  stated.  Problem  (6)  is  a 
special  case  of  problem  (a),  for  the  slope  at  a  point  on  a  curve 
really  shows  "  the  law  of  change  "  existing  between  the  ordinate 
and  the  abscissa  of  the  point  (see  Art.  2Q). 

A  brief  inspection  of  Arts.  24-26,  97,  99,  at  this  time,  will  repay 
the  beginner. 

Note.  Diflferential  calcnlns  and  integral  calculus.  The  subject  of 
infinitesimal  calculus  is  frequently  divided  into  two  parts ;  namely,  differential 
calculus  and  integral  calculus.  This  division  is  merely  a  formal  division  ; 
though  oftentimes  convenient,  it  is  by  no  means  necessary.  Examples  of  the 
kind  given  in  Arts.  2-4  formally  belong  to  "the  differential  calculus,"  and 
those  described  in  Arts.  5,  0,  to  "the  hitegral  calculus." 

7.  Elementary  notions  used  in  infinitesimal  calculus.  The  prob- 
lems used  in  Arts.  2-4  put  in  evidence  some  notions  and  methods, 
the  consideration  and  development  of  which  constitute  an  impor- 
tant part  of  infinitesimal  calculus.     These  notions  are  : 

(1)  The  notion  of  varying  quantities  which  may  approach  as 
near  to  zero  as  one  pleases,  such  as  M^  and  AiCi  in  the  last  stages 
of  the  solution  of  the  problems  in  Arts.  3  and  4. 

(2)  The  notion  of  a  varying  quantity,  such  as  — ^   in   Art.  3 

or  — ^  in  Art.  4  j,  which  approaches  a  fixed  number  when  Afj 

(or  Aa^i)  becomes  more  nearly  equal  to  zero,  and  approaches  in 
such  a  way  that  the  difference  between  the  varying  quantity  and 
the  fixed  number  can  be  made  to  become,  and  remain,  as  small  as 
one  pleases,  merely  by  decreasing  A^i  (or  Aa^i). 


12  INFINITESIMAL    CALCULUS.  [Ch.  I. 

The  infinitesimal  calculus  gives  mathematical  definiteness  and 
exactness  to  these  notions,  and  a  convenient  notation  has  been 
invented  for  dealing  with  them.  From  these  notions,  with  the 
help  of  this  notation,  it  has  developed  methods  and  obtained 
results  which  are  of  great  service  in  such  widely  separated  fields 
of  study  as  geometry,  astronomy,  physics,  mechanics,  geology, 
chemistry,  and  political  economy. 

A  review  of  certain  notions  of  algebra  is  not  only  highly  advan- 
tageous but  absolutely  necessary  for  a  satisfactory  understanding 
of  the  calculus  and  for  good  progress  in  its  study.  Accordingly, 
Chapter  II.  is  devoted  to  the  consideration  of  the  notions  of  a 
variable,  a  function,  a  limit,  and  continuity. 

Note.  Reference  for  collateral  reading.  Perry,  Calculus  for  Engi- 
neers, Preface,  and  Arts.  1-18. 


CHAPTER   II. 

ALGEBRAIC    NOTIONS   ^ATHICH   ARE    FREQUENTLY 
USED    IN   THE    CALCULUS. 

8.  Variables.  When  in  the  course  of  an  investigation  a  quan- 
tity can  take  different  values,  the  quantity  is  called  a  variable 
quantity,  or,  briefly,  a  variable.  For  instance,  in  the  example  in 
Art.  3,  the  distance  through  which  the  body  falls  and  its  speed 
both  vary  from  moment  to  moment,  and,  accordingly,  are  said  to 
be  variables.  Again,  if  the  x  in  the  expression  or  +  3  be  allowed 
to  take  various  values,  then  x  is  said  to  be  a  variable,  and  ar  -|-  3 
is  likewise  a  variable.  If  a  steamer  is  going  from  New  York  to 
Liverpool,  its  distance  from  either  port  is  a  variable. 

Note  1.  Numbers  and  their  graphical  representation.  The  measures 
of  quantities  are  indicated  by  means  of  numbers.  For  instance,  if  a  distance 
is  30  feet,  its  measure  (when  a  foot  is  taken  as  the  unit  of  measurement)  is 
30  ;  and  its  measure  is  360  when  an  inch  is  taken  as  the  unit.  When  a 
quantity^  varies,  the  number  which  indicates  its  measure  varies.  Numbers 
which  involve  V—  1  are  called  imaginary  numbers  ;  other  numbers  are  said 
to  be  real  numbers.*  The  (so-called)  real  numbers  can  be  represented 
graphically  on  a  straight  line  L'  OL  extending  to  an  infinite  distance  in  both 
directions  from  O.     Let  unity  be  represented  by  some  arbitrarily  chosen 

*  Real  numbers  are  divided  into  two  classes,  algebraic  numbers  and  tran- 
scendental numbers.  Every  (real)  root  of  an  algebraic  equation,  ax"  +  bx^-^ 
+  •'•  +  bx  +  m  =  0,  with  integral  coefficients  is  called  an  algebraic  (real) 
number.  These  numbers  include  integers,  irrational  numbers  such  as  V2 
and  v^3,  and  fractional  numbers  formed  from  integers  and  irrational 
numbers,  A  real  number  which  cannot  be  a  root  of  an  algebraic  equation  of 
the  form  described  is  called  a  transcendental  number.  A  well-known 
number  of  this  kind  is  tt,  the  ratio  of  the  circumference  of  a  circle  to  its 
diameter.  Transcendental  numbers  are  irrational.-  There  are  far  more 
transcendental  numbers  than  algebraic.  For  an  interesting  brief  element- 
ary discussion  on  transcendental  numbers  see  Klein,  Famous  Problems  in 
Elementary  Geometry  (Beman  and  Smith's  translation,  Ginn  &  Co.),  in 
particular,  pages  51—54. 

13 


14  INFINITESIMAL   CALCULUS.  [Ch.  II. 

length,  say  OM.  Let  the  distances  of  the  points  on  tlie  line  he  measured 
from  0,  and,  according  to  the  usual  convention^  let  the  distances  of  points 
on  the  right  of  0  be  regarded  as  positive  (and  be  given  a  plus  sign),  and  the 
distances  of  points  on  the  left  of  0  be  regarded  as  negative  (and  be  given  a 
minus  sign).     To  each  point  P  on  OL  there  corresponds  a  definite  number, 

— I ^ \ 1 ^ 

L!  0        M  PL 

Fig.  3. 

viz.  the  ratio  OP :  OM,  the  number  w  say  ;  and  to  each  number,  for  instance 
n,  there  corresponds  a  definite  point  P  such  that  0P=  n-  OM.  Positive 
numbers  are  represented  by  the  points  on  the  right  of  0,  and  negative 
numbers  by  the  points  on  the  left  of  0.  When  a  point  moves  along  the 
line  from  0  to  X,  it  passes  over  every  point  from  0  to  i  in  succession,  and 
represents  successively  each  number  from  zero  to  the  ratio  OL  :  OM.  Some 
of  these  numbers  are  integral,  such  as  1,  3,  12  ;  some  are  fractional,  such 
as  \,  f,  -i^ ;  and  some  are  incommensurable,  such  as  V^,  V8,  v'7,  tt.  The 
value  of  an  incommensurable  number  can  be  expressed  by  fractional  numbers 
to  as  close  an  approximation  to  exactness  as  one  pleases  ;  and  the  correspond- 
ing point  on  L'OL  can  be  located  as  nearly  to  absolute  correctness  as  one 
pleases.  For  instance,  V2  =  1.4142  •••  ;  accordingly,  the  point  corresponding 
to  \/2  lies  betv^een  the  points  corresponding  to  1.4  and  1.5;  betw^een  the 
points  corresponding  to  1.41  and  1.42  ;  between  the  points  corresponding 
to  1.414  and  1.415  ;  and  so  on. 

As  to  the  graphical  representation  of  imaginary  numbers  see  Chrystal, 
Algebra  (ed.  1886),  Part  I.,  Chap.  XII.,  §  2. 

In  this  course,  with  the  exception  of  a  few  instances,  only  real  numbers 
are  met. 

The  value  of  a  number  without  regard  to  sign  is  called  its  absolute  value. 
Thus  the  absolute  values  of  the  numbers  1,  —  2,  ^,  —  i  are  1,2,  i,  ^.  The 
absolute  value  of  a  number  x.  is  denoted  by  the  symbol  j  x  | . 

Note  2.  Infinite  numbers.  The  student  has  a  general  idea  of  the  set 
of  numbers  ordinarily  called  Jinite  numbers.  There  is  also  a  set  of  numbers 
each  of  whose  (absolute)  values  is  "greater  than  any  number  that  can  be 
named "  or  is  "beyond  all  bounds. "  These  numbers  are  said  to  be  infinitely 
great  numbers  or  infinite  numbers.  Finite  numbers  have  each  one  distinct 
symbol  at  least,  as  2,  \/2,  ],  |,  or  .4,  etc. ;  but  infinite  numbers  have  each 
the  same  symbol,  namely  oc,  which  is  called  "infinity." 

Instead,  however,  of  reading  x  =  cc,  "x  is  equal  to  infinity,"  it  is  better 
to  say  "  xis  infinitely  great,"*^  or  "  x  is  infinite,^''  or  "  x  is  beyond  all  bounds.'*'' 
The  phrase  "is  equal  to  infinity  "  may  give  the  impression  that  oo  denotes  a 
single,  definite,  immense  quantity  ;  an  impression  which  is  erroneous.  For 
instance,  consider  a  number  and  its  logarithm  to  base  10.  Log  10  =  1, 
log  100  =  2,  log  1000  =  3,  log  1,000,000  =  0,  log  1,000,000,000,000  =  12,  and 


9.]  FUNCTIONS.  15 

so  on.  It  is  evident  that  when  the  logarithm  is  infinitely  great,  the  corre- 
sponding number  is  also  infinitely  great.  Now  these  infinitely  gr.at  numbers 
are  very  different  from  each  other ;  for  when  the  logarithm  bt comes  infinite, 
the  corresponding  number  is  much  further  along  (so  to  say)  in  the  set  of 
infinite  numbers.  But  both  these  numbers  (the  logarithm  and  the  anti- 
logarithm)  are  then  denoted  by  the  same  symbol,  viz.  go.* 

Note  3.  References  for  collateral  reading  on  numbers.  Echols, 
Calculus,  Arts.  1-9  ;  Harkness  and  Morley,  I)itroductio?i  to  the  Theoi-y  of 
Analytic  Functions,  Chaps.  I.,  II. ;  Whittaker,  Modern  Analysis,  ('hap.  I. 

9.  Functions.  The  area  of  a  circle  varies  when  the  radius 
varies,  and  when  the  radius  has  a  definite  value,  the  area  has  a 
corresponding  definite  value.  The  volume  of  a  cube  varies  with 
its  length,  and  when  the  length  has  a  definite  value,  the  volume 
has  a  corresponding  definite  value.  To  the  sine  of  an  angle  there 
correspond  certain  definite  values  of  the  angle.  The  number  of 
deaths  per  year  in  a  city  depends,  in  some  measure,  upon  the 
number  of  people  in  it,  and  in  each  city  thei-e  is  a  definite  number 
of  deaths  per  year.  These  facts  illustrate  the  following  definition : 
When  two  variables  are  so  related  that  to  a  definite  value  of  one  of 
them  there  corresponds  a  definite  value  {or  values)  of  the  other,  the 
second  is  said  to  be  a  function  of  the  first.  The  first  is  sometimes 
called  the  argument  of  the  function. 

The  following  definition  of  a  function  may  also  be  used  :  When 
\\\o  variables  are  so  related  that  the  value  of  one  of  them  depends 
upon  the  value  of  the  other,  the  first  is  said  to  be  the  dependent 
variable  or  to  be  a  function  of  the  second,  and  the  second  is  said 
to  be  the  independent  variable  or  to  be  the  argument  of  the  func- 
tion. 

The  exact  relation  between  the  variables  may,  or  may  not,  be 
capable  of  definite  statement. 

If  ?/  =  2  x2  +  3  a:  -  7,  (1) 

the  value  of  y  varies  when  x  varies,  and  the  value  of  y  depends  upon  the  value 
of  x;  here  y  is  the  dependent  variable  (or  the  function),  and  x  is  the 
independent  variable  (or  the  argument).  On  the  other  hand,  the  value  of 
X  varies  when  y  varies,  and  the  value  of  x  depends  upon  the  value  of  y.     If 

*  The  two  infinitely  great  numbers  here  referred  to  are  compared  in 
Appendix,  Note  C  (Art.  3,  Ex.  1). 


16  INFINITESIMAL    CALCULUS.  [Ch.  II. 

y  be  allowed  to  vary,  x  must  vary  in  a  manner  to  suit ;  in  such  a  case  y  is  the 
independent  variable,  and  x  is  the  dependent  variable  (or  the  function). 
Precisely  the  same  remarks  may  be  made  if  x  and  y  are  connected  by  the 

relation 

x^y  +  y%  -f  a:2  _  3  y  +  7  =  0.  (2) 

When  a  relation  connecting  two  variables  is  given,  it  does  not 
matter,  except  in  so  far  as  convenience  is  concerned,  which  vari- 
able is  regarded  as  independent.  When  one  variable  is  chosen 
as  the  independent  variable,  the  other  must  be  considered  the 
function. 

The  value  of  a  variable  may  depend  upon  the  values  of  two  or 
more  variables,  or  it  may  have  a  definite  value  (or  several  definite 
values)  when  two  or  more  other  variables  have  definite  values. 
In  such  a  case  the  first  variable  is  said  to  be  a  function  of  the 
other  two,  or  the  first  variable  is  said  to  be  a  dependent  variable, 
and  the  other  two  are  said  to  be  independent  variables. 

Thus  the  distance  which  a  vessel  sails  from  a  port  depends 
both  upon  the  time  since  departure  and  upon  the  speed ;  in  other 
words,  the  distance  sailed  is  a  function  of  the  time  and  the  speed, 
^gain,  if  y2  j^Zz^-\-  x"  +  11  =  0, 

2;  is  a  function  of  x  and  y. 

Note.     On  the  term  "function,"  read  Gibson,  Calculus,  §  11. 

10.  Constants.  When  a  quantity  remains  unchanged  during  the 
course  of  an  investigation,  the  quantity  is  said  to  be  a  constant. 
Thus  in  the  case  of  a  steamer  going  from  New  York  to  Liverpool 
the  distance  of  the  steamer  from  either  port  is  variable,  and  the 
distance  between  the  ports  is  constant.  If  a  quantity  has  the 
same  value  in  every  investigation,  it  is  said  to  be  an  absolute 
constant;  if  it  has  a  particular  value  in  one  investigation,  and 
another  value  in  a  second  investigation,  and  so  on,  it  is  said  to 
be  an  arbitrary  constant. 

Thus  g  (Art.  3),  the  ratio  tt,  2,  |  are  absolute 
constants.  In  each  of  the  triangles  (Fig.  4)  having 
a  common  vertical  angle  a,  let  x  and  y  denote  the 
lengths  of  the  sides  containing  this  angle,  and  let 
A  denote  the  area  of  the  triangle.  Then,  by  trigo- 
nometry, A  =  \xy  sin  a.  Here  A^  x,  and  y  are 
yariables,  \  is  an  absolute  constant,  and  a  is  an  arbitrary  constant. 


10-12.]  CLASSIFICATION  OF  FUNCTIONS.  17 

11.  Classification  of  Functions. 

A.  Explicit  and  implicit  functions.  When  a  function  is  expressed 
directly  in  terms  of  the  dependent  variable,  like  y  in  equation  (1), 
Art.  9,  the  function  is  said  to  be  an  explicit  function.  When  the 
function  is  not  so  expressed,  as  in  equation  (2),  Art.  9,  it  is  said 
to  be  an  implicit  function.  If  relation  (2),  Art.  9,  were  solved  for 
y,  then  y  would  be  expressed  as  an  explicit  function  of  x. 

B.  Algebraic  and  transcendental  functions.  Functions  may  also 
be  classified  according  to  the  operations  involved  in  the  relation 
connecting  a  function  and  its  dependent  variable  (or  variables). 
When  the  relation  involves  only  a  finite  number  of  terms,  and 
the  variables  are  affected  only  by  the  operations  of  addition,  sub- 
traction, multiplication,  division,  raising  of  powers,  and  extraction 
of  roots,  the  function  is  said  to  be  algebraic;  in  all  other  cases 

it  is  said  to  be  transcendental.     Thus  2  x-  -{-3  x  —  7,  -y/x  -f  -,  are 

X 

algebraic  functions  of  x ;  sin  a',  tan  (x  +  a),  cos~^  x,  l"",  e^,  log  x, 
log  3  X,  are  transcendental  functions  of  x.  The  elementary  tran- 
scendental functions  are  the  trigonometric,  anti-trigonometnc,  expo- 
nential, and  logarithmic.     Examples  of  these  have  just  been  given. 

C.  Continuous  and  discontinuous  functions.  A  discussion  on  this 
exceedingly  important  classification  of  functions  is  contained  in 
Art.  16. 

12.  Notation.  In  general  discussions  variables  are  usually 
denoted  by  the  last  letters  of  the  alphabet,  x,  y,  z,  u,  v,  •••,  and 
constants  by  the  first  letters,  a,  b,  c,  •••. 

The  mere  fact  that  a  quantity  is  a  function  of  a  single  variable, 
X,  say,  is  indicated  by  writing  the  function  in  one  of  the  forms 
/(a?),  F(x),  <f>(x),  •",fi{x),f{x),  •••.  If  one  of  these  occurs  alone, 
it  is  read  "  a  function  of  x  "  or  "  some  function  of  x  "  ;  if  several 
are  together,  they  are  read  "  the  /function  of  a;,"  "  the  i^-f unction 
of  a;,"  "the  phi-function  of  a;,"  •••.  The  letter  y  is  often  used  to 
denote  a  function  of  a;. 

The  fact  that  a  quantity  is  a  function  of  several  variables, 
X,  y,  z,  •••,  say,  is  indicated  by  denoting  the  quantity  by  means  of 
some  one  of  the  symbols,  f(x,  y),  <f>(x,  y),  F(x,  y,  z),  if/(x,  y,z,u),'". 
These  are  read  "the  /-function  of  x  and  ?/,"  "the  phi-function 
of  X  and  2/,"  "  the  i^-f unction  of  x,  y,  and  z,^'  etc. 


18  INFINITESIMAL    CALCULUS,  [Ch.  II. 

Sometimes  the  exact  relation  between  the  function  and  the 
dependent  variable  (or  variables)  is  stated ;  as,  for  example, 

f(x)  =  aj-+  3x  —  7,ovy  =  x^-\- 3 a?  —  7 ;  F(x,  y)=2 e^+  7  e«-{-xy  -  1. 

In  such  cases  the  /-function  of  any  other  number  is  obtained  by 
substituting  this  number  for  x  in  f(x),  and  the  i^-function  of  any 
two  numbers  is  obtained  by  substituting  them  for  x  and  y  respvic- 
tively  in  F(x,  y).     Thus 

/(^)  =  ^2  +  3  ^  -  7,  /(4)  =  42  +  3  .  4  -  7  =  2 1 ; 

F(t,  z)=2e'+7e'-i-tz-l,  F(2,  3)  =  2  e^  +  7  e^  +  5. 

EXAMPLES. 

1.  Calculate  /(2)  and  /(.I)  when  f{x)=  3\/^+- +  7  a;2  +  2.  Write 
/(^),  /(w),/(sinx).  ^ 

2.  Calculate  /(2,  3),  /(-2,  1),  and  /(-I,  -1)  when  f{x,  y)  = 
3  x2  +  4  xi/  +  7  1/2  -  13  X  +  2  ^  -  11.     Write  /(w,  v),  /(sin  x,  2). 

2  -I-  3  a; 

3.  Calculate  2:  as  a  function  of  x  when  y  =f(x)  =  —^ and  z  =  f(y). 

4  —  7  X 

4.  Given  that  /(x)=  x^  +  2  and  F(x)  =4  +  Vx,  calculate  /[i^(x)]  and 

5.  If  /(x,  y^  =  ax2  +  6x|/  +  cy^  write  /(i/,  x),  /(x,  x),  and  /(?/,  y). 

6.  If  2/  =  /(x)  =  ^^Jl^,  show  that  x  =/(«/). 

ex  —  a 

2  X  —  1 

7.  If   y-=(/)Cx)  = ,    show  that  x  =  0(^),  and  that  x=z<p^(x),  in 

3x  —  2 

which  <f)^(x)  is  used  to  denote  0[0(x)]. 

8.  If/(x)  =^-i-l,  show  that  /2(x)  =  x,  p(x)  =  x,  /^(x)  =  x,  etc.,  in 
which  /2(x)  is  used  to  denote  /[/(x)],  /^(x)  to  denote  /{/[fix)]},  etc. 

9.  If /(:«)  =^^,  show  that    /(^)-/Cy^    =^_ll^. 

^  +  1  l+/(x)-/(2/)      l  +  x?/ 

Note.  Notation  for  inverse  functions.  The  student  is  already  familiar 
with  the  trigonometric  functions  and  their  inverse  functions,  and  with  the 
notation  employed  ;  thus,  y  =  tan  x,  and  x  =  tan-i  y.  In  general  if  ?/  is  a 
function  of  x,  say  y  =/(x),  then  x  is  a  function  of  y.  The  latter  is  often 
expressed  thus  :  x  =f~^  (y).  For  instance,  \i  y  =  logx,  x  =  log-i  (^).  This 
notation  was  explained  in  England  first  by  J.  F.  W.  Herschell  in  1813,  and  at 
an  earlier  date  in  Germany  by  an  analyst  named  Burmann.  See  Hersrhell, 
A  Collection  of  Examples  of  the  Application  of  the  Calculus  of  Finite 
Differences  (Cambridge,  1820),  page  5,  Note. 


13.]  BEPRESENTATION  OF  FUNCTIONS.  19 

13.   Geometrical  representation  of  functions  of  one  variable.*    The 

fact  that  the  relation  between  a  function  and  its  independent 
variable  can  be  made  manifest  to  the  eye  by 
means  of  a  curve,  is  familiar  to  students  of 
analytic  geometry.  For  instance,  let  y=2x-\-Sj 
and  draw  a  line  MN  having  a  slope  2  and  an 
intercept  3  on  the  ?/-axis.  The  line  3IN  may 
be  regarded  as  a  picture  or  geometrical  repre- 
sentation of  the  function  2  ic  +  3.  For  any 
value  of  X  the  length  of  the  ordinate  drawn  from  the  corresponding 
point  on  the  x-axis  to  the  line  MN  will  give  the  value  of  the 
function  2  a;  -|-  3. 

How  to  draw  the  curve  whose  equation  is  y=f(x),  or,  tchat  is 
the  same  thing,  how  to  picture  or  represent  the  function  f(x),  or 
make  a  graph  of  the  function  /(a?),  is  one  of  the  fundamental 
problems  in  analytic  geometry.  For  instance,  if  the  function  y 
and  the  independent  variable  x  have  the  relation  x^  -{-  ^  =  25, 
then  the  curve  which  represents  the  function  y,  i.e.  V2o  —  a^,  is 
the  circle  whose  centre  is  at  the  origin  of  coordinates  and  whose 
radius  is  5  units  in  length. 

EXAMPLES. 

1.  Draw  the  curve  which  represents  the  function  x^x.  (That  is,  make 
the  graph  of  Vx,  or  draw  the  curve  whose  equation  is  y  =  y/x.) 

2.  Draw  the  curves  which  represent  the  following  functions,  and  write  the 
equations  of  the  curves  : 

(a)   Sx-5,  (b)    |x  +  3,  (c)    V49  -  x^,  (d)   4x2, 

(e)    Vx2  -  49,         (/)   sinx,  (g)   cosx,  (h)   tan  x. 

On  the  one  hand,  some  properties  of  the  graph  of  a  function  can 
he  predicted  by  means  of  what  is  termed  "a  discussion  of  the 
equation  '^  involving  the  function  and  its  dependent  variable. 
For  instance,  if  x^  +  y^  =  16,  it  is  obvious  that  for  any  value  of  x 
there  are  two  values  of  y  which  are  numerically  equal  but  opposite 

*  References  for  collateral  reading  and  review  on  this  topic:  Hall, 
Introduction  to  Graphical  Algebra;  Chrystal,  Introduction  to  Algebra, 
Chaps,  v.,  XXV.;  Gibson,  Calculus,  Chaps.  IT.,  III.  ;  Tanner  and  Allen, 
Analytic  Geometry,  Chap.  III.  and  Art.  49 ;  and  other  texts  on  algebra 
and  on  analytic  geometry. 


20  INFINITESIMAL   CALCULUS.  [Ch.  II. 

in  sign;  hence  the  graph  of  y  must  be  symmetrical  about  the 
i»-axis.  It  is  also  evident  that  y  has  real  values  when  x  is 
between  —  4  and  +  4,  and  that  y  is  zero  when  ic  is  —  4  or  +  4, 
and  that  y  is  imaginary  when  x  is  less  than  —4  and  greater 
than  +4.  Hence  the  graph  of  y  must  lie  between  the  vertical 
lines  drawn  at  x  =  —  4  and  «  =  4. 

On  the  other  hand,  important  properties  of  a  function  can  he 
discovered  by  an  inspection  of  its  representative  curve*  or  graph. 
Thus,  in  Arts.  63,  64,  74,  76,  etc.,  graphs  are  used  in  the  investi- 
gation of  functions ;  especially  because  these  curves  tend  to 
make  the  investigations  simpler  and  clearer  for  beginners.  These 
investigations  of  functions  can  be  conducted,  however,  without 
any  reference  to  representative  curves.  But  geometrical  repre- 
sentation of  functions  serves  to  illustrate  and  emphasise  properties 
already  known  about  the  functions,  and  also  serves  as  a  means 
for  the  discovery  of  new  properties. 

Note.  The  geometrical  representative  of  a  function  of  two  variables  is  a 
surface.  Thus,  \i  z  =  y/cfi  —  x"^  —  2/"^,  the  geometrical  representative  of  z  is 
a  sphere  of  radius  a  with  its  centre  at  the  origin  of  coordinates.  See 
Art.  79,  Note  1. 

14.  Limits.  The  notion  that  varying  quantities  may  have  j^o^er? 
limiting  values  is  very  important,  and  should  be  clearly  understood 
before  entering  upon  the  study  of  the  calculus. 

EXAMPLES. 

1.    The  number  0.3333  ••• ,  i.e.  the  sum  of  the  geometric  series 

varies  with  ?i,  the  number  of  terms  in  the  series.  The  greater  n  is,  the  more 
nearly  does  the  sum  of  this  series  come  to  \.  If  w  is  a  million,  the  sum  of 
the  series  is  very  near  to  i ;  if  n  is  a  million  million,  the  sum  is  still  nearer 
to  \.  But  the  sum  of  the  series,  even  if  the  number  of  terms  be  infinite 
{i.e.  greater  than  any  number  that  can  be  named),  can  never  be  actually  \. 
Nevertheless,  if  any  positive  number^  say  e,  no  matter  how  near  to  zero  but 
not  actually  zero,  be  assigned.,  it  is  possible  to  take  a  number  of  terms,  n  say., 

*  It  is  not  the  case  that  every  function  of  one  variable  can  be  represented 
by  a  curve.  The  question,  what  are  the  circumstances  under  which  it  is 
impossible  for  a  graph  of  a  function  to  be  drawn,  will  not  be  discussed  here. 
(See  Harnack,  Calculus,  Art.  15.)  All  the  functions  which  the  student  will 
meet  in  this  course  can  be  represented  by  curves. 


14.]  EXAMPLES   ON  LIMITS.  21 

of  (1),  such  that  the  difference  between  the  sura  of  these  n  terms  and  I  will 
be  less  than  the  assigned  number  e,  and  will  remain  less  than  e  for  any 
greater  number  of  terms.  (Thus,  if  e  be  .000001,  then  n  can  be  6  or  any- 
greater  number.)  This  is  expressed  mathematically  by  saying  "  the  limiting 
value  (or,  briefly,  '  the  limit ')  of  the  sum  of  the  series  -^^  -\-  yf^  +  j-^^-^  +  ••• , 
as  the  number  of  terras  approaches  an  infinitely  great  number,  is  |." 

2.  In  Fig.  1,  Art.  4,  let  Q  move  along  the  curve  until  it  comes  to  P. 
When  Q  moves  toward  P,  PB  or  Ax  becomes  smaller.  If  any  length  be 
assigned  (say  .00001  inch),  then  Q  can  move  so  near  to  P  that  Aa;  shall 
become  and  remain  less  than  .00001  inch.  Finally,  when  Q  reaches  P,  Ao; 
becomes  zero.  All  this  is  expressed  in  mathematical  language  by  saying 
"  the  limit  of  Ax,  when  Q  approaches  P,  is  zero."  Similarly,  the  limit  of  the 
chord  PQ,  as  the  arc  QP  approaches  zero,  is  zero. 

3.  In  Fig.  1,  Art.  4,  when  Q  moves  toward  P  the  angle  PGX  ap- 
proaches the  angle  PWX.  If  any  angle  be  assigned,  say  e",  §  can  be  made 
to  approach  so  near  to  P  that  the  difference  between  the  angles  PGX  and 
PWX  will  be  less  than  e",  and  will  continue  to  be  less  than  e"  when  Q 
approaches  still  nearer  to  P.  Finally,  when  Q  reaches  P,  PGX  becomes 
PWX.  This  is  expressed  in  mathematical  terms  by  saying  "  the  limit  of  the 
angle  PGX.,  when  Q  approaches  P,  is  PWX.''"' 

4.  Show  that  in  Fig.  1,  Art.  4,  "the  limit  of  the  angle  TPQ.,  as  the 
arc  PQ  approaches  zero,  is  zero." 

5.  Let  a  regular  polygon  of  n  sides  be  inscribed  in  a  circle.  When  the 
number  of  sides  is  increased  the  length  of  the  polygon  becomes  more  and 
more  nearly  equal  to  the  length  of  the  circle.  These  lengths  can  never 
become  exactly  equal  ;  but  the  difference  between  them  can  be  made  less 
than  any  positive  number  that  may  be  assigned,  simply  by  increasing 
the  number  of  the  sides  ;  and  this  difference  will  continue  to  remain  less 
than  the  assigned  number  when  the  number  of  the  sid  j  is  further  increased. 
This  is  expressed  mathematically  thus:  "The  limit  of  the  length  of  the 
perimeter  of  a  regular  polygon  inscribed  in  a  circle,  as  the  number  of  sides 
approaches  an  infinitely  great  number,  is  the  length  of  the  circle." 

6.  Show  that  "the  limit  of  the  area  of  a  regular  polygon  inscribed  in  a 
circle,  as  the  number  of  sides  approaches  an  infinite  number,  is  the  area 
of  the  circle." 

7.  Enunciate  and  prove  propositions  similar  to  Exs.  6,  6,  about  a  circle 
and  a  circumscribing  regular  polygon. 

8.  The  number   varies   with   n.      As   n  increases  this   nuraber 

(-2)« 
decreases  and  approaches  nearer  and  nearer  to  zero.      It  can  never  reach 
zero.     But  by  increasing  n  the  difference  between  the  number  and  zero  can 
be   made    less    than    any    positive    number  that    may   be    assigned  ;    and 
on  further  increasing  n  this  difference  will  continue  to  be  less  than  the 

assigned  number.     Accordingly,  the  limit  of  the  variable  nuraber  , 

as  n  approaches  an  infinitely  great  value,  is  zero.  ^~    ^ 


22  INFINITESIMAL    CALCULUS.  [Ch.  II. 

9.    Show  that  the  limit  of  — ,  as  n  approaches  an  infinite  number,  is 
zero. 

(The  number  in  Ex.  8  is  alternately  positive  and  negative  according  as  n 
is  even  or  odd  ;  hence,  it  is  alternately  greater  and  less  than  its  limit.  The 
number  in  Ex.  9  is  always  positive,  and,  accordingly,  is  always  greater  than 
its  limit.) 

10.  Show  that  the  limit  of  the  sum  2  —  1  +  |  —  •••  to  n,  terms,  as  n 
increases  beyond  all  bounds,  is  |. 

11.  In  Ex.  (a),  Art.  4,   — ^  varies  with  Ax,  and  approaches  4  as  Ao; 

approaches  zero.    By  decreasing  Ax  the  difference  between  — ^  and  4  can  be 

made  less  than  any  positive  number  that  may  be  assigned,  and  will  remain  less 

than  this  number  when  Ax  continues  to  decrease.     That  is,  the  limit  of  -^, 
as  Ax  approaches  zero,  is  4. 

Show  that  in  Ex.  (6),  Art.  4,  the  limit  of  — ^,  as  Ax  approaches  zero,  is  2  x. 

Ax 

Note  1.    In  each  of  these  cases  — ^  finally  reaches  its  limit.     In  Ex.  10  the 

Ax 

variable  sum  can  never  reach  its  limit. 

As 

12.  In   Ex.   (6),  Art.  3,    —    varies   with   A^,  and  approaches  gt  as  A« 

approaches  zero.    By  decreasing  A«  the  difference  between  —  and  qt  can  be 

A^ 

made  less  than  any  positive  number  that  may  be  assigned,  and  will  remain 

less  than  this  number  when  A^  continues  to  decrease.     Accordingly,  the  limit 

of  — ,  as  AJ  approaches  zero,  is  gt. 
At 

As 
In  Ex.  (a).  Art.  3,  the  limit  of  — ,  as  At  approaches  zero,  is  128.8. 

As  ^^ 

In  each  of  these  cases  —  can  reach  its  limit. 
At 

13.  (a)  Show  that  the  limit  of  ?1I1_,  as  6  approaches  zero,  is  1. 

6 

(&)    Show  that  the  limit  of  -ML-,  as  6  approaches  zero,  is  1. 
6 

(c)  Show  that  the  limit  of  cos  6,  as  6  approaches  zero,  is  1. 

(d)  Show  that  the  limit  of  sin  6,  as  6  approaches  zero,  is  0. 

(e)  Show  that  the  limit  of  sin  6,  as  d  approaches  -,  is  1. 

7-2  _  n"^ 

14.  Show  that  the  limit  of —,  when  x  approaches  a,  is  2  a. 

X  —  a 

Note  2.     The  limit  of  a  constant  is  the  constant  itself. 


14.]  DEFINITION  OF  A    LIMIT.  23 

Definition  of  a  limit.  Let  there  be  a  function  of  a  variable,  and 
let  the  variable  approach  a  particular  value.  If  at  the  same  time 
as  the  variable  approaches  the  particular  value,  the  function  also 
approaches  a  fixed  constant  in  such  a  way  that  the  absolute  value 
of  the  difference  between  the  function  and  the  constant  may  be  made 
less  than  any  positive  number  that  may  be  assigned ;  and  if 
moreover,  this  differeiice  continues  to  remain  less  than  the  assigned 
number  when  the  variable  approaches  still  nearer  to  the  particidar 
value  chosen  for  it;  then  tlie  constant  is  the  limit  of  the  function  when 
the  variable  approaches  the  particular  ralue. 

Note  3.    This  definition  may  be  expressed  in  a  slightly  different  form,  viz. : 

Let  the  variable  x  approach  a  particular  value  a  ;  if  (1)  as  a:  approaches 
nearer  to  a,f(x)  approaches  nearer  to  a  fixed  constant  A,  and  if  (2)  a  number, 
say  €,  being  taken  as  small  as  one  pleases,  zero  excepted,  it  is  possible  to  find  a 
number  h  such  that  |  f(a  -\-  h)  —  A\<C\e\,  and  if  (3)  as  h  decreases,  the  quan- 
tity \f(a  -\-  h)  —  A\  continues  to  be  less  than  e  ;  then  A  is  said  to  be  a  limit 
of  f{x)  as  X  approaches  a. 

Note  4.  If  the  difference  between  the  varying  function  and  the  fixed 
constant  can  actually  become  zero,  the  limit  is  an  attainable  limit;  if  the 
difference  can  never  be  zero,  the  limit  is  an  unattainable  limit. 

Ex.  Mention  the  variables  and  functions  in  Exs.  1-14  above,  which  have 
attainable  limits. 

Note  5.  A  variable  or  a  function  may  be  always  less  than  its  limit,  or 
always  greater  than  its  limit,  or  sometimes  greater  and  sometimes  less  than 
its  limit. 

Ex.  Examine  the  variables  and  functions  in  Exs.  1-14  with  respect  to 
this  matter. 

Note  6.  The  importance  of  the  phrase  '■'-and  continues  to  remain  less''^ 
in  the  definition  of  a  limit  should  be  clearly  apprehended.  A  variable 
function  may  approach  a  constant  value  by  a  series  of  advances  and  retreats. 
Thus  a  point  may  move  on  th3  line  OX  from  A  to  31  in  the  following  way. 

O  A  B  A.     AiAi     J.3  A5  M  X 

Fig.  6. 

It  may  go  forward  from  A  to  ^1,  then  back  to  A^.,  then  forward  to  Az,  then 
back  to  Ai,  then  forward  to  A5,  and  so  on.  While  on  the  whole  it  is  getting 
nearer  to  31,  still  its  distance  from  31  does  not  continue  to  remain  less  than 
an  assigned  value.     For  instance,  after  it  arrives  at  Az  it  goes  back  to  A^. 


24  INFINITESIMAL   CALCULUS.  [Ch.  II. 

The  idea  of  a  limit,  which  has  already  been  applied  practically 
by  the  student  in  arithmetic,  algebra,  geometr}^,  and  trigonometry, 
plays  a  very  great  and  important  part  in  calculus. 

15.  Notation.  The  limit  of  a  variable  quantity,  and  the  con- 
dition under  which  this  limit  is  approached,  are  expressed  by 
means  of  a  certain  mathematical  shorthand.  Thus  the  last  sen- 
tence in  Ex.  1,  Art.  14,  is  expressed  : 

Lim,^^  (fV  +  -rh  +  To'o  0  +  -•)  =  i- 
The  result  found  in  Ex.  11  is  expressed: 

The  result  found  in  Ex.  13  (e)  is  expressed : 
Lim^^l  sin  6  =  1. 

The  symbol  =  is  placed  between  a  variable  and  a  constant  in 
order  to  indicate  that  the  variable  approaches  the  constant  as  a 

limit.     Thus  0  =  ^  above,  means  that  6  approaches  ^  as  a  limit. 

Note.  The  symbol  =  is  used  to  indicate  an  approach  to  equality.  The 
symbol  =  is  used  by  many  instead  of  =  to  indicate  the  same  idea.  Various 
other  notations  are  also  employed.* 

Ex.  Express  the  results  in  Exs.  1-14  in  the  mathematical  manner  of 
writing. 

16.  Continuous  variables.  Continuous  functions.  Discontinuous 
functions.  A  number  x  is  said  to  vary  continuously  from  the  value 
a  to  the  value  b  when  in  changing  froyn  a  to  b  it  takes  each  value 
between  a  and  b  once  and  only  once. 

Thus  (Fig.  6)  let  OA  =  a,  OB  =  &,  and  x  denote  the  distance  from  0  of 
any  point  on  OX.  If  a  point  moves  along  OXfrom  A  to  B,  then  x  varies  con- 
tinuously from  a  to  b,  and  is  said  to  be  a  continuous  variable.  The  distance 
through  which  a  body  falls,  varies  continuously  from  when  the  body  starts 


*  Professor  Echols  of  the  University  of  Virginia  advocates  the  use  of  the 
symbol  £  for  the  term  "limit"  and  the  symbol  (  =  )  in  preference  to  =.  See 
his  Calculus  (Holt  &  Co.),  preface  and  Arts.  12,  13. 


15,  16.]  CONTINUOUS  FUNCTIONS,  25 

until  it  stops,  and  accordingly  is  a  continuous  variable.  Again,  if  a  point 
move  along  a  circle,  both  the  arc  and  the  chord  measured  from  a  fixed  point 
on  the  circle  to  the  moving  point  vary  continuously,  i.e.  are  continuous 
variables.  In  the  case  of  a  steamer  going  straight  ahead  from  New  York  to 
Liverpool  without  stopping  its  distance  from  New  York  is  a  continuous 
variable  ;  so  also  is  its  distance  from  Liverpool. 

A  function  f(x)  is  said  to  be  a  continnons  function  of  x  for  all 
values  of  x  from  x  =  a  to  x  =  b,  when  it  satisfies  the  following 
conditions : 

(1)  Its  absolute  value  does  not  become  infinite  for  any  value  of 
X  between  a  and  b ; 

(2)  The  corresponding  change  in  f{x)  is  also  infinitely  small 
when  an  infinitely  small  change  is  made  in  x  while  the  value  of  x 
lies  between  a  and  b.  In  other  words,  if  the  value  of /(a-)  does  not 
take  a  sudden  jump  of  either  a  finite  or  infinite  amount  when  x 
changes  by  only  an  infinitely  small  amount  at  any  value  between 
a  and  b. 

Note  1.  The  second  condition  is  expressed  more  formally  and  rigorously 
in  Note  7.  The  notion  of  a  continuous  function  becomes  clearer,  and  is 
developed  more  fully,  by  means  of  investigations  in  the  calculus. 

Ex.  1.    Let  f{x)  =  x2  +  3  X  -  7. 

This  function  does  not  become  infinite  for  any  finite  value  of  x  ;  accord- 
ingly, condition  (1)  is  satisfied.  Let  Xi  be  any  finite  value  of  a;,  and  h  be  an 
infinitely  small  change  which  is  made  in  x  when  x  =  x\.     Then 

Kxi)  =  a;i2  +  3  5^1  -  7,  and  /(.ri  +  h)  =  {x^  +  h^  +  ^{x^^h)-  7.   ,^ 

...  /(a;i  +  h)-f(ixi)=h(2xi  +h  +  3). 

The  first  member  is  the  change  in  the  function  corresponding  to  the 
infinitely  small  change  h  in  x  when  x  =  X\.  The  second  member  can  evi- 
dently be  made  as  near  zero  as  one  pleases  simply  by  decreasing  h,  and  thus 
is  infinitely  small ;  accordingly  /(a;)  satisfies  condition  (2).  Thus  /(x) 
satisfies  both  conditions  (1)  and  (2),  and,  accordingly,  is  a  continuous 
function  of  x  for  all  finite  values  of  x. 

If  in  the  case  of  a  function  f(x),  either  of  the  conditions  (1)  and 
(2)  is  not  fulfilled  when  x  has  a  particidar  value,  say  x  =  c,  then  the 
function  f(x)  is  said  to  be  discontinuous  for  the  value  x  =  c,  or,  more 
briefly,  discontinicous  at  c. 


26  INFINITESIMAL    CALCULUS.  [Ch.  It.  . 

Ex.  2.  Let  f{x)  =  tanx.  (Here  x  denotes  the  number  of  radians  in  the 
angle. ) 

This  function  is  continuous  from  x  =  0^  to  x  =—,  and  is  continuous  in 

o 
many  other  intervals.     But  tan  —  is  infinite,  and  accordingly,  tan  x  is  dis- 
continuous for  x=-' 
2 
The  change  that  tan  x  makes  when  x  passes  through  the  value  ~,  should 

also  be  noted.     Let  h  denote  a  quantity  exceedingly  near  to  zero. 

Then  tan  (  -  —  /t  ]  =  an  exceedingly  great  positive  quantity, 

and  tanf  -  +  /i  j  =  an  exceedingly  great  negative  quantity. 

. ••  tan  (  -  —  ^  J  —  tan (~-\-h\  =  the  sum  of  two  exceedingly  great  positive 
^  '  ^  '  quantities. 

Thus,  when  x  makes  an  exceedingly  small  change  in  passing  from  one  side 
of  the  value  -  to  the  other,  the  function  tanx  makes  a  tremendous  jump 

and  changes  by  an  exceedingly  great  amount.     This  is  also  apparent  from 
an  inspection  of  the  representative  curve  of  y  =  tan  x. 

Note  2.     When  x=^,  tanx  may  be  either  an  infinitely  great  positive 

quantity  or  an  infinitely  great  negative  quantity.     If  x  is  increasing  from 

0^  to  — ,  then,  when  x  reaches  the  value  -,  tanx  has  an  infinitely  great  posi- 

tive  value  ;  but  if  x  is  decreasing  from  tt  to  -,  then,  when  x  reaches  the  value 

-,  tanx  has  an  infinitely  great  negative  value.      Thus  the  value  of  tanx 

for  X  =  -,  depends  upon  the  way  in  which  x  has  approached  to  the  value  — 
2  ^ 

Note  3.  An  instance  of  a  function  which  takes  a  sudden  Jinite  jump  is 
given  in  Ex.  3. 

Note  4.  In  a  first  course  in  calculus  the  student  will  meet,  in  general, 
with  functions  only  when  they  are  continuous.  The  remainder  of  this  article 
is  not  absolutely  necessary  for  simple  work  in  calculus ;  but  it  may  interest 
a  beginner,  and  will  show  him  the  necessity  there  is  for  distinguishing  func- 
tions as  continuous  and  discontinuous. 

1  1 

Ex.  3.    Examine  the  function  /(x)  =  2(4*-^  -  1)  -=-  (4*-3  +  1)  when  x  =  8. 
Let  A  be  a  quantity  as  near  zero  as  one  pleases. 

1-1  1-1 

(.1)  1  1  I 

4^  *  —  1      ^  4*  4*  —  1       «         4* 

4^^  h)^l  -  +  1  ik^i  1  +  ^ 

4A  4A 


16.] 


DISCONTINUOUS  FUNCTIONS. 


27 


When  h  approaches  zero,  -  approaches  an  infinite  value ;  then  4*  ap- 
proaches an  infinite  value,  and  —  approaches  zero.  Accordingly,  when  h  is 
very  nearly  zero  ^a 

/(3  -Ji)  =  -2  nearly,  and  f(S-\-  h)  =  +  2  nearly. 

Therefore,  when  h  is  very  small,  /(3  +  ^)-/(3  -  h)  is  exceedingly  near 
to  4.  Accordingly,  when  x  in  changing  passes  through  the  value  3,  f(x) 
changes  by  the  amount  4  ;  and  thus  /(x)  is  discontinuous  for  x  =  3. 

Note  5.  When  (Ex.  3)  a:  =  3,  /(x)  may  be  either  +  2  or  —  2.  If  x  is 
increasing  from  0  toward  3,  then,  when  x  =  3,  /(x)  =  —  2  ;  but  if  z  is  de- 
creasing from  4  toward  3,  then,  when  x  =  3,  /(x)  =  +  2.  Thus  the  value  of 
/(x)  for  X  =  3  depends  upon  the  loay  in  which  x  has  approached  the  value  3. 


The  representative  curve  is  shown  in  Fig.  7.  When  x  is  moving  toward  the 
right  the  ordinate  PM  of  the  curve  arrives  at  the  line  AB  (the  line  x  =  3), 
with  the  value  —  2,  and  leaves  this  line  with  the  value  -f  2.  When  x  is 
moving  toward  the  left  the  ordinate  PiMi  of  the  curve  arrives  at  AB  with 
the  value  +  2,  and  leaves  this  line  with  the  value  —  2. 

Note  6.  A  simple  instance  of  a  discontinuous  function  in  nature.  The 
temperature  of  a  body  (whether  solid,  liquid,  or  gaseous)  subjected  to  heat 
depends  upon  the  quantity  of  heat  which  has  been  received  (or  absorbed)  by 
the  body.  When  heat  is  added  to  a  solid  body  the  temperature  of  the  body 
rises.  If  heat  is  added  continuously,  for  a  time  the  temperature  continues  to 
rise.  But  whenever  the  body  begins  to  take  the  liquid  form  ("to  melt"), 
the  temperature  becomes  stationary.,  and  it  remains  stationary  (while  heat  is 
being  added)  until  the  body  is  completely  liquefied.  Then  the  temperature 
begins  to  rise  again,  and  continues  to  rise  until  the  liquid  begins  to  vaporize, 
when  it  again  becomes  stationary.  The  temperature  remains  stationary 
(while  heat  is  being  added)  until  the  liquid  has  all  been  vaporized,  when  it 
again  begins  to  rise,  and  continues  to  rise  so  long  as  the  gas  continues  ,to 
receive  heat.  Thus  in  the  case  of  a  mass  of  matter  subjected  to  heat  the  tem- 
perature of  the  matter,  in  general,  is  a  continuous  function  of  the  heat  which 
is  absorbed.     But  there  are  two  "breaks"  in  the  continuity  ;  these  occur 


28  INFINITESIMAL    CALCULUS.  [Ch.  11. 

when  the  matter  is  melting  and  when  it  is  changing  into  the  gaseous  form. 
The  matter  arrives  at  tlie  "  melting  point,"  or  temperature  of  fusion,  possessed 
of  a  certain  amount  of  heat,  and  leaves  that  temperature  possessed  of  an 
additional  amount  of  heat;*  it  arrives  at  the  "boiling  point,"  or  tempera- 
ture of  vaporisation,  possessed  of  a  certain  amount  of  heat,  and  leaves  that 
temperature  possessed  of  an  additional  amount  of  heat,  t 

Ex.  Make  a  sketch  to  illustrate  this  case.  Lay  off  on  a  horizontal 
axis  the  successive  temperatures  of  the  body,  and  on  a  vertical  axis  the 
amounts  of  heat  received  by  the  body. 

Ex.  4.    Show  that  ?/  is  a  continuous  function  of  x  when  x'^  -{-  y^  =  16. 

Ex.  5.    Show  that  the  function  -  is  discontinuous  only  when  x  is  zero. 

X 

Find  the  change  in  the  function  when  x  passes  through  zero  from  a  negative 
value  to  a  positive  value,  and  find  the  change  when  x  passes  through  zero 
from  a  positive  to  a  negative  value.     The  curve  ?/  =  -  is  an  hyperbola  whose 

X 

branches  are  in  the  first  and  third  quadrants,  and  whose  asymptotes  are  the 
axes  of  coordinates. 

Ex.  6.    Given  that  ^ =  5^-^,  show  that  the  function  y  is  discontinuous 

2/  -  2 

for  ic  =  1.     Show  that,  for  x  increasing,  when  x  reaches  the  value  1,  y  has  the 

value  1,  and  when  x  leaves  the  value  1,  y  has  the  value  2. 

Ex.  7.   Examine  the  function  y  at  its  point  of  discontinuity  when 

gx  -  a  \ 

y  =  a — J ,  inwjiiche>l. 

e^  4-  1 

Note  7,  More  formal  (and  more  rigorous)  definitions  of  continuous 
functions  are  the  following  : 

A.  A  function  f{x)  is  said  to  be  continuous  throughout  the  interval  from 
x  =  a  to  X  =  b,  when 

(1)  /(x)  does  not  become  infinite  for  any  value  of  x  between  a  and  b  ;  and 

(2)  at  any  point  in  this  interval,  as  Xi,  it  is  always  possible  to  find  a  value 
of  h  for  which  J  /(xi  +  h)  —f{x{)  \  is  less  than  any  number  as  small  as  one 
pleases,  say  e,  that  may  be  assigned. 

B.  A  function  f(x)  is  said  to  be  continuous  when  x  =  a^\i 

(1)  /(«)  is  not  infinite  and  has  a  definite  value  (or  definite  values). 

(2)  limit,i„/(x)=/(a). 

An  inspection  of  the  definition  of  a  limit,  Note  3,  Art.  14,  and  a  comparison 
of  definitions  A  and  JS,  show  that  conditions  (2)  are  practically  identical. 

*  "Latent  heat  of  fusion."     (See  text-books  on  Physics.) 
t  "  Latent  heat  of  vaporisation. " 


16.]  CONTINUITY.  29 

Note  8.  The  following  important  proposition  can  be  deduced  from  the 
definitions  of  continuity  in  Note  7,  viz. :  If  a  function  f(x)  is  everywhere 
continuous  in  the  interval  from  x  =  a  to  x  =  b,  and  xi^  x-y  are  two  points  in 
this  interval,  then  as  x  goes  through  the  range  of  values  from  x\  to  X2  the 
function  assumes,  once  at  least,  each  value  which  lies  between  /(x'l)  and 
f(x-2).  In  other  words,  the  continuous  function  does  not  overleap  any  values 
intermediate  between  two  values  which  it  assumes.  The  meaning  of  this 
proposition  will  be  made  clearer  by  a  reference  to  Figs.  5,  20  a,  &,  c.  In 
Fig.  5,  when  x  varies  from  2  to  3,  y  takes  all  values  from  7  to  9. 

Note  9.  References  for  collateral  reading.  On  Limits  2iXi^  Continuo^is 
and  Discontinuous  Functions  :  Chrystal,  Algebra  (Ed.  1889),  Part  I., 
Chap.  XV.,  Part  II.,  Chap.  XXV.  (in  particular  §§1-13,  24,  26)  ;  Harkness 
and  Morley,  Introduction  to  the  Theory  of  Analytic  Functions,  Chap.  VI. 
(VII.);  Lamb,  Calculus,  Chap.  I.;  Gibson,  Calculus,  Chaps.  IV.,  V.  ; 
Harnack,  Calculus  (Cathcart's  translation),  §§  9-19 ;  Echols,  Calculus, 
Arts.  12-25.  Also  see  references  for  collateral  reading,  Art.  21.  AVith  refer- 
ence to  the  topic  in  Note  8,  see  Whittaker,  Modern  Analysis,  Art.  30. 


CHAPTER   III. 

INFINITESIMALS,    DERIVATIVES,    DIFFERENTIALS, 
ANTI-DERIVATIVES,    AND   ANTI-DIFFERENTIALS. 

17.  In  this  chapter  some  of  the  principal  terms  used  in  the 
calculus  are  defined  and  discussed,  and  one  of  the  main  problems 
of  the  calculus  is  described.  In  the  first  study  of  the  calculus 
it  is  better,  perhaps,  not  to  read  all  this  chapter  very  closely, 
but  after  a  cursory  reading  of  it  to  proceed  to  Chapter  IV.,  and, 
while  working  the  examples  in  that  chapter,  to  re-read  carefully 
the  articles  of  this  chapter.  These  articles  can  also  be  reviewed 
most  profitably  when  the  special  problems  to  which  they  are 
applied  are  taken  up.  Articles  22,  23,  however,  should  be  care- 
fully studied  before  Chapter  IV.  is  begun. 

18.  Infinitesimals,  infinite  numbers,  finite  numbers.  An  injini- 
tesimal  is  a  variable  which  has  zero  for  its  limit.  (See  definition 
of  a  limit.  Art.  14.)     That  is,  if  a  denote  an  infinitesimal, 

a  =  0,    or    limit  a  =r  0. 

For  instance,  in  Ex.  (a),  Art.  4,  when  PR  is  approaching  zero  it 
is  an  infinitesimal.  So  also,  at  the  same  time,  are  angle  QPT 
and  the  triangle  PQR.  Again,  when  angle  0  is  an  infinitesimal 
sin  6  and  tan  6  are  infinitesimal ;  cos  6  is  an  infinitesimal  when  0 

is  approaching  ^ ;  when  n  is  increasing  beyond  all  bounds  1-5-2" 

is  an  infinitesimal. 

Note.  The  infinitesimal  of  the  calculus  Is  not  the  same  as  the  infinitesimal 
of  ordinary  speech.  The  latter  is  popularly  defined  as  "  an  exceedingly  small 
quantity,"  and  is  usually  understood  to  have  a  fixed  value.  The  infinitesimal 
of  the  calculus,  on  the  other  hand,  is  a  variable  which  approaches  zero  in  a 
particular  way. 


17-19.]  INFINITESIMALS.  31 

The  following  statements  are  in  accordance  with,  or  follow 
directly  from,  the  definitions  of  a  limit  and  an  infinitesimal. 

(1)  The  difference  between  a  variable  and  its  limit  is  an 
infinitesimal.  That  is,  on  denoting  the  variable  by  x  and  the 
limit  by  a, 

if  limit  X  =^a,  i.e.  if  x  =  a, 

then  x  =  a-\-  aj    in  which  a  =  0. 

(2)  If  the  difference  between  a  constant  and  a  variable  is  an 
infinitesimal,  then  the  constant  is  the  limit  of  the  variable.     In 


"J"^"^^"?  ..-.. 

x=^a-\ 

-a, 

in  which 

«  =  0, 

then 

x  =  a, 

i.e. 

limit 

x  =  a. 

This  principle  has  been  employed  in  the  exercises  in  Arts.  3,  4. 

It  is  evident  that  the  reciprocal  of  an  infinitesimal  approaches 
a  number  which  is  greater  than  any  number  that  can  be  named, 
namely,  an  infinite  number.  Accordingly,  an  infinite  number  may 
be  defined  as  the  reciprocal  of  an  infinitesimal.  Numbers  which 
are  neither  infinitesimal  nor  infinite  are  called  finite  numbers. 

19.  Orders  of  magnitude.  Orders  of  infinitesimals.  Orders  of 
infinites.  Let  m  and  n  each  denote  a  number  which  may  be 
finite,  infinite,  or  infinitesimal.     When  the  limiting  value  of  the 

ratio  —   is  a  finite  number,  m  and  n  are  said  to  be  finite  with 
n 

respect  to  each  other  and  to  be  of  the  same  order  of  magnitude; 

when  the  limit  of  the  ratio  —  is  either  zero  or  infinity,  m  and  n 

n 
are  said  to  be  of  different  orders  of  magnitude. 

For  instance,  1,897,000,000  and  .000001  are  of  the  same  order  of  magni- 
tude. Tan  90°  and  tan  45°  are  of  different  orders  of  magnitude.  Log  x 
and  X  are  of  different  orders  of  magnitude  when  x  is  an  infinite  number. 
(See  Appendix,  Note  C,  Art.  3,  Ex.  1.) 

That  infinitesimals  may  be  of  different  orders  of  magnitude  is 
shown  by  the  following  illustration. 


32 


INFINITESIMAL    CAL CUL  US. 


[Ch.  III. 


Suppose  that  the  edge  BL  of  the  cube  in  Fig.  8  is  divided  into  any  number 
of  parts,  and  that  each  part,  as  Bb,  becomes  infinitesimal.     Through  each 


point  of  division,  as 


6,  let  planes  be  passed  at  right  angles  to  BL.  The 
cube  is  thereby  divided  into  an  infinite  number  of 
infinitesimal  slices  like  Bd.  Now  suppose  that  the 
edge  BA  is  divided  like  BL  into  parts  like  Bf  which 
become  infinitesimal,  and  let  a  plane  be  passed 
through  each  point  of  division  /  at  right  angles  to  BA. 
The  slice  Bd  is  thereby  divided  into  an  infinite  num- 
ber of  infinitesimal  parallelopipeds  like  Ck.  Finally 
suppose  that  the  edge  BC  is  divided  into  parts  which 
become  infinitesimal  like  Bg,  and  that  through  each 
point  of  division,  as  g,  a  plane  is  passed  at  right 
angles  to  BC.  Then  Ck  is  thereby  divided  into  an 
infinite  number  of  infinitesimal  parallelopipeds  like 

^,    ^\   ^,  is  infinite, 
Bd     Ck     kg 


kg.     Since  the  limiting  value  of  each  of  the  ratios 


the  parallelopipeds  DL,  Bd,  Ck,  kg,  are  all  of  different  orders  of  magnitude. 

Deflnition.     If  a  is  an  infinitesimal,  and  ^  is  such  that  the 

limit  of  the  ratio  —  is  a  finite  number,  then  (3  is  said  to  be  an 
a 

infinitesimal  of  the  same  order  of  magnitude  as  a,  and  ^  is  said 

to  be  finite  with  respect  to  a.     If  ^  is  such  that  the  limit  of  the 

ratio  — ,  in  which  n  is  a  positive  integer,  is  finite,  then  fi  is  said 
a" 

to  be  an  infinitesimal  of  the  nth  order  with  respect  to  a. 

In  order  to  determine  the  orders  of  infinitesimals,  it  is  neces- 
sary to  take  some  one  infinitesimal  as  a  standard  infinitesimal, 
and  this  standard  infinitesimal  is  said  to  be  of  the  first  order. 
If  the  standard  infinitesimal  be  denoted  by  a,  then  «-,  a?,  •••,  a", 
are  said  to  be  infinitesimals  of  the  second,  third,  •••,  nth  orders, 
respectively. 

Infinite  numbers,  being  reciprocals  of  infinitesimals,  also  have 
different  orders  of  magnitude.     With  reference  to  the  standard 

infinitesimal  «,  -  is  an  infinitely  great  number.     The  numbers 

1111^ 

-,  — ,  — ,  •••,  —  (i.e.  a~^,  a~^,  •••,  <«"**),  are  said  to  be  infinites  of 

If  a  number  ^ 


a    a"    0^'  a 

the   first,  second,  •••,  nth  orders,  respectively 

be  such  that  the  limiting  value  of  the  ratio  yS-r--  (i.e.  I3a~^)  is  a 

a 


10.]  THEOREMS  ON  INFINITESIMALS.  33 

finite  number,  then  ^  is  said  to  be  an  infinite  of  the  first  order, 
and  /3  and  a~'^  are  said  to  be  finite  with  respect  to  each  other. 
If  the  limit  oi  ^  -. —  (i.e.  ^a'"")  is  finite,  then  /3  is  said  to  be  an 
infinite  of  order  n. 

Theorems  on  inflnitesimals.  (a)  The  product  of  an  infinitesimal 
a,  and  any  finite  number  k,  namely  Ka,  is  an  infinitesimal  of  the 
same  order  as  a.     This  follows  at  once  from  the  definition  above.* 

CoR.  1.  The  sum  of  a  finite  number  of  infinitesimals  of  the 
same  order  is  an  infinitesimal  of  that  order. 

CoR.  2.  The  algebraic  sum  of  a  finite  number  of  infinitesimals 
is  an  infinitesimal,  t 

(b)  The  product  of  two  infinitesimals,  ^  and  y  say,  of  orders 
m  and  n  respectively,  is  an  infinitesimal  of  order  m  4-  n.  For,  if 
a  denote  the  standard  infinitesimal,  ft  =  kiu"*,  y  =  Ksa",  and  hence 
fty  =  Ki/f2a'"+'*,  which  is  an  infinitesimal  of  order  m  +  n.  (Here 
K,  and  K2  are  finite  numbers.) 

(c)  The  quotient  ft  -i-  y  (see  (6))  is  an  infinitesimal  of  order 
m  —  n. 

N.B.  These  theorems  are  true  for  numbers  of  any  magnitude, 
for  finite  and  infinite  numbers  as  well  as  for  infinitesimals.  The 
student  can  make  the  proofs. 

EXAMPLES. 

1.  Let  (Fig.  8)  AB  =  1,  and  let  Bg,  Bf,  fk,  be  infinitesimals  of  the  first 
order,  (i)  Show  that  the  volumes  of  Bd,  Ck,  and  kg  are  infinitesimals  of  the 
first,  second,  and  third  orders  respectively,  with  respect  to  the  volume  of  DL. 
(ii)  Show  that,  with  respect  to  Ck,  DL  and  Bd  are  infinites  of  the  second 
and  first  orders  respectively,  and  kg  is  an  infinitesimal  of  the  first  order, 
(iii)  Show  that,  with  respect  to  kg,  the  volumes  of  Ck,  Bd,  and  DL  are 
infinites  of  the  first,  second,  and  third  orders  respectively. 

*The  product  of  an  infinitesimal  and  an  infinite  number  may  be  infini- 
tesimal, finite,  or  infinite,  according  to  circumstances.  Particular  instances 
are  given  in  Appendix,  Note  C. 

t  The  limiting  value  of  the  sum  of  an  infinite  number  of  infinitesimals 
may  be  infinitesimal,  finite,  or  infinite,  according  to  circumstances.  For 
simple  illustrations  see  McMahon  and  Snyder,  Diff.  Cal,  page  12.  Many 
instances  in  which  this  limiting  value  is  finite  will  be  found  later  in  this  book. 


34  INFINITESIMAL   CALCULUS.  [Ch.  III. 

2.  Show  that,  if  Aic  in  Fig.  1  be  an  infinitesimal  of  the  first  order,  then 
A«/  is  an  infinitesimal  of  the  first  order,  and  the  area  of  triangle  PQB  is  an 
infinitesimal  of  the  second  order. 

3.  Show  that  if  angle  d  be  an  infinitesimal,  sin  d  and  tan  d  are  infini- 
tesimals of  the  same  order  as  d.  (See  Ex.  13,  Art.  14,  and  Plane  Trigo- 
nometry, Art.  83.)     This  is  a  very  important  case  in  infinitesimals. 

4.  Let  d  denote  one  of  the  angles  of  a  right-angled  triangle,  x  the 
adjacent  side,  y  the  opposite  side,  and  r  the  hypothenuse.  Show  that  if  6  is 
an  infinitesimal  of  the  first  order,  r  and  x  are  both  finite,  or  both  infini- 
tesimals, or  infinites  of  the  same  order  ;  and  show  that  if  r  is  also  an  infinitesi- 
mal, y  is  an  infinitesimal  of  an  order  one  higher ;  and  if  r  is  an  infinite,  y  is 
an  infinite  of  an  order  one  less ;  and  if  r  is  finite,  y  is  an  infinitesimal  of  the 
first  order. 

5.  In  the  triangle  in  Ex.  4,  in  which  6  is  an  infinitesimal  of  the 
first  order,  show  that  if  r  be  an  infinitesimal  of  order  n,  r  —  x  is  of  order 
w  +  2. 

»-2sin2^  n 


fSu 


GGESTiON  :  r^  —  x"^  —  y"^  =  r^sin^  d  ;   whence  r  —  x 


r  -\-  X 


6.  In  a  circle  of  finite  radius  the  difference  between  the  length  of  an 
infinitesimal  arc  of  the  first  order  and  its  chord  is  an  infinitesimal  of  at 
least  the  third  order. 

Let  AB  be  an  arc  of  a  circle  of  finite  radius  r  and  centre  0.  Draw 
the  chord  AB  and  the  tangents  at  A  and  B. 
Thesa  tangents  meet  at  T;  OT  bisects  arc  AB, 
the  chord  AB,  and  the  angle  AOB.  Let  angle 
AOC=  6',  take  6  for  the  standard  infinitesimal. 
Then 

arc  AC  =rd  (trigonometry),  an  infinitesimal  of 
first  order,  and 

AM  =  r  sin  6,  an  infinitesimal  of  first  order 
(Ex.  3)  ;  also 

AT  =  rt3ind,  an  infinitesimal  of  first  order. 

Now  angle  MA T  =  d.  Hence,  by  Ex.  5,  AT  -  AM  is  an  infinitesimal  of 
the  third  order.  But  (SiTC  AC- AM)<(AT- AM).  Hence,2(arc^C-^itf), 
i.e.  arc  AB  —  chord  AB,  is  an  infinitesimal  of  at  least  the  third  order. 

Note.  The  theorem  stated  in  Ex.  6  holds  for  any  curve  of  finite  curvature. 
(See  McMahon  and  Snyder,  Diff.  Cal.,  Th.  4,  page  27  ;  also  see  Byerly, 
Diff.  Cal.,  Art.  165.) 


20.]  THEOEEMS  ON  LIMITS.  35 

20.  Theorems  on  limits  and  infinitesimals,  (a)  If  two  variables j 
X  and  y,  be  always  equal,  and  if  one  of  them,  say  x,  approaches  a 
limit  a,  then  the  other  approaches  the  same  limit;  that  is, 

if  ^  =  2/)  and  x  =  a,  then  y  =  a. 

For  x  =  a  -{-a,      in  which     a  =  0 ; 

hence  ?/  =  a  +  a ;       that  is       y  =  a. 

Ex.  1.  The  two  members  of  Equation  (3),  Art.  3  (ft),  are  always  equal, 
and  the  second  member  approaches  a  limit  gh  when  A^i  approaches  zero ; 
hence  the  first  member  approaches  the  same  limit. 

Ex.  2.  The  two  members  of  Equation  (2),  Art.  4  (6),  are  always  equal, 
and  the  second  member  approaches  a  limit  2  xi  when  AiCi  approaches  zero ; 
hence  the  first  member  approaches  the  same  limit. 

(6)  The  limit  of  the  sum  of  a  finite  number  of  variables,  x,  y,  z,  •••,  is 
equal  to  the  sum  of  their  limits. 

For,  if  x  =  a,  y  =  6,  •••, 

then  x  =  a-\-a,  y  =  b-\-p,  •••, 

in  which  a  =  0,  /3  =  0  •••. 

Hence  x  +  y  -\-  -"  =  a  +  b  +  -'  +  (a  +  P  +  — )• 

But  a  +  /3  +  •••  i  0.     [Art.  19,  Theorem  (a),  Cor,  2  ] 

Hence  lim  (a;  +  y  +  •••)  =  a  +  6  +  ••• 

=  lim  X  +  lim  y  +  •■•. 
Ex.      lim  .333  ...  =  lim  (^'^  +  j^^  +  ...)  =  ^,  lim  .141414  ...  =  ^f, 
lim  (.333  ...  +  .141414  ...)  =  lim  (.474747  -..) 
—  9?  —  I  +  i?- 

(c)  The  limit  of  the  product  of  a  finite  number  of  variables  is  the 
product  of  their  limits. 

For,  if  a;  =  a,  !/  =  6,  «  =  c, 

then  X  =  a  -\-  a,  y  =  b  -\-  p,  z  =  c  +  y, 

in  which  a  =  0,  /3  =  0,  7  i^.  0. 

Hence        xyz  =  abc  +  bca  +  ca/3  +  aby  +  apy  -\-  bya  +  ca/3  +  apy. 

.'.  lim  xyz  =  abc       [Art.  19,  Theorem  (a),  Cor.  2,  Art.  20,  Theorem  (a)] 

=  lim  x  •  lim  y  •  lim  z. 

H^.B.  Theorems  (a),  (b),  (c),  are  true  if  one  or  more  of  the  variables  be 
replaced  by  constants. 

Ex.  lim  (.333  ...  x  .141414  ..-)=  ^  •  i|  =  ^S. 


36  INFINITESIMAL   CALCULUS.  [Ch.  III. 

(d)    The  limit  of  the  quotient  of  two  variables,  x  and  y,  whose  limits  are 
finite,  is  the  quotient  of  their  limits. 

Since         x  =-  >  y,  lim x  =  lim  (  -  )  •  lim  y  by  Theorem  (c). 

y  \yJ 

x\  _lima; 
y)      lim  2/' 

Ex.  lim  •^3^-        '^■^•"       --      - 


.1414  ...      lim. 1414...      "       ^"       '* 

(e)  The  order  of  an  infinitesimal  is  not  altered  by  adding  or  subtracting 
another  infinitesimal  of  higher  order. 

Let  a  be  the  standard  infinitesimal,  and  ^  and  y  be  infinitesimals  of  orders 
m  and  n  respectively,  and  n  be  greater  than  m.     Now 

^  +  7^  ^    I    7    . 


hence 


lim  t±y.  :^  lim  A  +  lim  -^.  [Theorem  (6)] 


But  lim  -^  ==  0, 

a™ 

since  7  -=-  a*"  is  an  infinitesimal  of  order  n  —  w,  and  n  is  greater  than  m. 
...lim^^^^^lim-^. 

Note.  The  order  of  an  infinitesimal  is  not  altered  by  adding  an  infini- 
tesimal of  the  same  order,  but  it  may  be  altered  by  subtracting  one  of  the 
same  order.  E.g.  if  jS  =  2  a^  +  3  a*,  7  =  2  a^  -  3  a^,  5  =  a^  -  a^,  then  ^  +  7 
=  4  a3  +  3  a*  -  3  a^,  which  is  of  the  third  order  ;  /3  -  7  =  3  a*  +  3  a^,  which 
is  of  the  fourth  order  ;  /3  —  5  =  a^  +  3.a*  +  a^,  which  is  of  the  third  order. 

Ex.  Show  that  if  x  and  y  are  two  variables,  and  limit  (x  -^  y)  =  1,  then 
X  —  y  is  infinitesimal  with  respect  to  both  x  and  y. 

21.   Fundamental  theorems  of  the  calculus. 

A.  The  limit  of  the  quotient  of  any  two  variables,  x  and  y,  is  not 
altered  by  adding  to  them  any  two  numbers,  say  a  and  p,  which  are 
infinitesimal  to  x  and  y  respectively; 

that  is,  lim  ^^  =  lim  ^,  when  ^  =  0  and  ?  =  0. 

^  y  +  ^         y  ^  y 


l  +  « 


For 


X  -\-  a _x X 


21.]  FUNDAMENTAL   THEOllEMS.  '  37 

T      X  +  a      V      X    ■,-             X      T      X 
.'.  lim  — ■ —  =  lim  -  .  lim =  lim  -•     . 

y+^       y       i+i?        y 
y 

[Art.  20,  Theorems  (a),  (c),  (cZ).] 

Note  1.  This  is  sometimes  called  the  fundamental  theorem  of  the 
differential  calculus,  as  it  is  frequently  used  in  that  branch  of  the  infini- 
tesimal calculus.     See  Art.  22,  Notes  1,  2. 

Ex.  1.  Lim,.,2aM:J^e  ^  2^3  ^ 

Ex.  2.  Lim^^^o,  A^ol^T^  =  -*  t^ee  Ex.  3  {a) ,  Art.  4.  ] 

2  ?/  +  Ay      y 

B.  If  the  limit  of  the  sum  of  any  number  of  infinitesimals  of  the 
same  sign  be  finite,  this  limit  is  not  altered  when  any  infinitesimal 
is  replaced  by  another  the  limit  of  whose  ratio  to  the  first  infini- 
tesimal is  unity. 

Let  there  be  a  set  of  any  number  of  infinitesimals,  aj,  as,  •••,  a,,, 
whose  sum  approaches  a  finite  limit  as  n  becomes  infinitely  great. 
Let  /?!,  /Sgj  •••>  Pnj  ^6  another  set  of  infinitesimals,  such  that 

liin-  =  l,  lira^'  =  l,   ...,  lim  — =  1.  (1) 

According  to  (1), 

in  which  ^\  =  ^,  ^9  =  0,  •••,  e„  =  0. 

Then     ^^=za^-\-  Cjaj,  ^^  =  <^2  +  f-^^-i,   '",  /?«  =  ««  +  c«a„ ; 

and    ^1  +  A  H h  /8„  =  aj  +  ao  H h  a„  +(ciai  +  Cgao  H h  c„a„). 

•••    (A  +  )S2  +  •••  +  A.)  -  (ai  +  a,  4-  •••  +  «„)  =  £,«!  +  c^ao  +  •••  +  c„a„. 
.-.  lim  (/?!  +  ^2  4-  ...  +  ^,,,)  _  lim  (a^  +  ag  +  •••  +  «,) 

=  lim  (citti  +  €2^2  H h  e„a„). 

But  lim  (Citti  H-  £202  +   •  •  •   +  Cnttn)   =  0- 

Por  let  r)  be  the  numerically  greatest  of  the  e's,  then 

(ejaj  -f  egog  -f-  t,.  -f-  €„a^)  <;  (^aj  -f  (Xg  +  ' "  +  a„)?y. 


38  INFINITESIMAL   CALCULUS.  [Ch.  III. 

Now    lim  (ttj  +  a2  +  •  •  •  +  a„)    is    finite,    by    hypothesis,    and 
lim^  =  0;   hence        ihn  (a, +  -.+  -+ a^r;  =  0, 

and  accordingly,     lim  (cjai  +  caas  + h  e„a„)  =  0. 

Hence     lim  (^i  +  i^,  +  -  +  /8„)  =  lim  (a^  +  a^  +  -  +  a„). 

Note  2.     This  is  sometimes  called  the  fundamental  theorem  of  the  integral 
calculus,  as  it  is  often  used  in  that  branch  of  infinitesimal  calculus. 

Note  3.     A  simple  proof  of  B,  depending  on  a  theorem  on  fractions,  is 
given  in  Gibson's  Calculus,  page  198. 

Note   4.      References    for    collateral    reading    on    infinitesimals. 

McMahon  and  Snyder,  Differential  Calculus  (American  Book  Co.),  Arts. 
1-15;  J.  J.  Hardy,  Infinitesimals  and  Limits  (Chemical  Publishing  Co., 
Easton,  Pa.),  pamphlet  22  pages  ;  Gibson,  Elementary  Treatise  on  the 
Calculus,  Arts.  86,  87  ;  Byerly,  Diff.  Cal.,  Chap.  X. 

22.   The  derivative  of  a  function  of  one  variable.    Suppose  that 
the  function  .-/  ") 

denotes  a  continuous  function  of  x.  Let  x  receive  an  increment 
Aa; ;  then  the  function  becomes 

f{x  -h  ^x).  (a) 

Hence  the  corresponding  increment  of  the  function  is 

f{x-^^x)-f{x).  (b) 

This  may  be  written  A  [/(o^)]. 

The  ratio  of  this-  increment  of  the  function  to  the  increment 
of  the  variable  is 

f(x  +  Ax)-f(x),       .^    Ar/(.T)1  .. 

Ax  '  Ax  ^  ^ 

TJie  limit  of  this  ratio  when  Ax  approaches  zero,  i.e. 

T  f(x-\-Ax)  —  f(x)  1-  A  fix)  ... 

Ax  Ax 

is  called  the  derived  fnnction  of  /(«)  with  respect  to  x ;  or  the 
derivative  (or  the  derivate)  of  f{x)  with  respect  to  x-,  or  the 
a;-derivative  of /(a?).  It  is  also  called  the  differential  coefficient 
of  f(x),  a  name  which  is  explained  in  Art.  27. 


22.]  DIFFERENTIATION.  39 

If  y  also  be  used  to  denote  the  function,  that  is,  if 

then  if  x  receive  an  increment  Aa:,  y  will  receive  a  corresponding  increment 
(positive  or  negative),  which  may  be  denoted  by  Ay,  i.e. 

y  +  Ay  =f{x  +  Ax). 

Hence  Ay  =f(x  +  Ax)  —  /(x)  ; 

and  '         Ay^/(^  +  Ax)-/W. 

,  Aa;  Aa;  ^  ^ 

Ax  Ax 

The  process  of  finding  the  derivative  of  a  function  is  called 
differentiation.  This  process  is  a  perfectly  general  one,  as  indi- 
cated in  steps  (a),  (6),  (c),  and  (d).  It  may  be  described  in 
words,  thus : 

(1)  Give  the  independent  variable  an  increment; 

(2)  Find  the  corresponding  increment  of  the  function ; 

(3)  AYrite  the  ratio  of  the  increment  of  the  function  to  the 
increment  of  the  variable. 

(4)  Find  the  limit  of  this  ratio  as  the  increment  of  the  variable 
approaches  zero. 

For  a  slightly  different  description  of  the  process  of  differentiation,  see 
Note  4. 

Note  1.  To  differentiate  a  function  (i.e.  to  find  its  derivative)  is  one 
of  the  three  main  problems  of  the  infinitesimal  calculus,  and  is  the  main 
problem  of  that  branch  which  is  called  "  the  differential  calculus.''^ 

Note  2.  The  other  two  main  problems  of  the  infinitesimal  calculus  (see 
Arts.  27  rt,  94)  are  the  main  problems  of  that  branch  called  "  the  integral 
calculus.^''  It  may  be  said  here  that  while  the  differential  calculus  solves  the 
proUem,  "  when  the  function  is  given,  to  find  the  derivative,"  on  the  other 
hand  the  integral  calculus  solves  as  one  of  its  two  main  problems  the  inverse 
problem,  namely,  "when  the  derivative  is  given,  to  find  the  function." 

EXAMPLES. 

1.   Find  the  derivative  of  x^  with  respect  to  x. 

Here  f(x)  =  x^  (See  Fig.,  p.4^2.) 

Let  X  receive  an  increment  Ax  ; 
then  /(x  +  Ax)  =  (x  +  Ax)^  =  x-"^  +  3  x^Ax  +  3  x(Ax)2  +  (Ax)3. 


40  INFINITESIMAL   CALCULUS.  [Ch.  III. 

.-.  fix  +  ^x)  -  fix)  =  3  x^Ax  +  3  xiAxy  +  (Axy. 

Ax 

.-.  11111^.^0-^^^  +  ^-^^—-^^^^-  =  3  x2. 

Ax 

If  y  be  used  to  denote  the  function,  thus  y  =  x^,  then  the  first  members  of 

these  equations  will  be  successively,  y,  y  +  Ay,  Ay,  — ^,  lim^^io— • 

Ax  Ax 

Note  3,  It  should  be  observed  that  the  expression  (c)  depends  both  on 
the  value  of  x  and  the  value  of  Ax,  and,  in  general,  contains  terms  that 
vanish  with  Ax,  as  exemplified  in  Ex.  1.  (This  is  shown  clearly  in  Art.  176.) 
On  the  other  hand,  the  value  of  the  derivative  depends  on  the  value  which 
X  has  when  it  receives  the  increment,  and  on  that  alone.  For  this  reason,  the 
derivative  of  a  function  is  often  called  the  derived  function.  For  instance, 
in  Ex.  1,  if  X  =  2,  the  value  of  the  derivative  is  12  ;  if  x  :=  6,  the  value  of 
the  derivative  is  108.  Compare  Exs.  in  Arts.  3,  4.  (It  is  probably  now 
apparent  to  the  beginner  that  the  process  used  in  the  problems  in  Arts.  3,  4, 
was  nothing  more  or  less  than  differentiation.) 

Note  4.  Sometimes  Ax  is  called  the  difference  of  the  variable  x,  (&)  is 
called  the  corresponding  difference  of  the  function,  and  (c)  is  called  the 
difference-quotient  of  the  function.  The  process  of  differentiation  may  then 
be  described,  thus :  (1)  Make  a  difference  in  the  independent  variable  ; 
(2)  Calculate  the  corresponding  difference  made  in  the  function  ;  (3)  Write 
the  ratio  of  the  difference  in  the  function  to  the  difference  in  the  variable  ; 
(4)  Determine  the  limiting  value  of  this  ratio  when  the  difference  in  the 
variable  approaches  zero  as  a  limit. 

2.  Find  the  derivatives,  with  respect  to  x,  of  x,  2  x,  3  x,  ax,  x^,  7  x^, 
11  x^,  6x2,  yfi^  5  yfi^  13  gj3^  and  cx^. 

Ans.  1,  2,  3,  a,  2x,  14  x,  22  x,  2  &x,  3x2,  15x2,  39x2,  ^cx\ 

3.  Calculate  the  values  of  these  functions  and  the  values  of  their 
derivatives,  when  x=:l,  x  =  2,  x  =  3. 

4.  Find  the  derivatives,  with  respect  to  x,  of :  («)  x2  +  2,  x2  —  7, 
x2  +  ^• ;  (6)  x^  +  7,  x3  -  9,  x3  +  c. 

1    2 

5.  Differentiate  x*,  x2  +  4  x  —  5,   -,  -  —  3  x  +  2  x2,  with  respect  to  x. 

XX  o 

6.  Find  the  derivatives,  with  respect  to  t,  of  3  «2,  4  «3  _  g  «  +  -• 

3  7  ^ 

7.  Differentiate  %^,  -  y^  —  S  y  —  -,  with  respect  to  ?/. 

4  y 

8.  Show  that,  if  n  is  a  positive  integer,  the  derivative  of  a?»»  with  respect 
to  X,  is  nic»»-i. 

Note  5.  The  result  in  M,  8,  ^^  wiU  be  seen  l^^er,  iq  ^r^e  for  qU  con- 
stant values  of  n,  , 


23.]  NOTATION.  41 

9.   Assuming  the  result  in  Ex.  8,  apply  it  to  solve  Exs.  4-7. 

Note  6.  In  order  that  a  function  may  be  differentiable  (i.e.  have  a  deriva- 
tive), it  must  be  continuous ;  all  continuous  functions,  however,  are  not 
differentiable.  For  remarks  on  this  topic,  see  Echols,  Calculus,  Art.  30. 
For  an  example  of  a  continuous  function  which  has  nowhere  a  determinate 
derivative,  see  Echols,  Calculus,  Appendix,  Note  1,  or  Harkness  and  Morley, 
Theory  of  Functions,  §  65. 

23.  Notation.  There  are  various  ways  of  indicating  the  deriva- 
tive of  a  function  of  a  single  variable.  (In  what  follows,  the 
independent  variable  is  denoted  by  x.  In  the  case  of  other 
variables  the  symbols  are  similar  to  those  now  to  be  described 
for  functions  of  x.) 

(a)  This  symbol  is  often  used  to  denote  (d)  Art.  22,  viz. 

/'(^).  A 

Thus  the  derivatives  (or  derived  functions)  of  F{x),  <f>(y),  f{t), 
fi(z),  with  respect  to  x,  y,' t,  and  z,  respectively,  are  denoted  by 
F'(x)j  <l>'(y),  f'(t),  fi(z).  These  are  sometimes  read  "  the F-prime 
function  of  .t,"  etc. 

(b)  If  y  is  used  to  denote  the  function  of  x  (see  Art.  22),  the 
derivative  of  y  with  respect  to  x  is  frequently  indicated  by  the 
symbol  ^,  ^ 

This  is  often  read  "  i/-prime  " ;  but  it  is  better  to  say  "  deriva- 
tive of  2/." 

(c)  The  a^derivative  of  f(x)  is  also  indicated  by  the  symbol 

The  brackets  in  D  are  usually  omitted,  and  the  symbol  is  written 

df(x) 


doc 


E 


Symbols  C,  D,  and  E  should  be  read  "the  ic-derivative  of /(a;)." 
{d)  When  y  denotes  the  function,  the  derivative  (see  Equation 
(/)  Art.  22)  is  sometimes  denoted  by 


42  INFINITESIMAL   CALCULUS.  [Ch.  III. 

The  brackets  in  F  and  G  are  usually  omitted,  and  the  symbol 
for  the  derivative  is  written 

%  H 

dx 

This  should  be  read  for  a  while  at  least  by  beginners,  "the 
derivative  of  y  with  respect  to  ic,"  or  more  briefly  "  the  x-derivative 
ofy"  (Other  phrases,  e.g.  "  dy  by  dx/'  are  common,  but,  unfortu- 
nately, are  misleading.) 

(e)  In  case  (d)  the  operation  of  differentiation,  and  also  its 
result,  namely,  the  derivative,  are  alike  indicated  by  the  symbol 

(/)  Sometimes  the  independent  variable  x  is  shown  in  the 
symbol,  thus  2)  ?/.  r 

Note  1.  Mathematics  deals  with  various  notions,  and  it  discusses  these 
notions  in  a  language  of  its  own.  In  the  study  of  any  branch  of  mathe- 
matics, the  student  has  Jirst  to  clearly  understand  its  fundamental  notions, 
and  then  to  learn  the  peculiar  .shorthand  language,  made  up  of  signs  and 
symbols  and  phrases,  which  has  been  in  part  invented,  and  in  part  adapted, 
by  mathematicians.  A  striking  instance  of  the  great  importance  of  mere 
notation  is  seen  in  arithmetic.  To-day  a  young  pupil  can  easily  perform, 
arithmetical  operations  which  would  have  taxed  the  powers  of  the  great 
Greek  mathematicians.  The  one  enjoys  the  advantage  of  the  convenient 
Arabic  notation*  for  numerals,  the  other  was  hampered  by  the  clumsy 
notation  of  the  Greeks. 

Note  2.  Symbols  A  and  B,  and  also  /and  J,  have  this  important  quality, 
namely,  they  tend  to  make  manifest  the  fact  that  the  derivative  is  a  single 
quantity.  It  is  not  the  ratio  of  two  things,  but  is  the  limiting  value  of  a 
variable  ratio.     Symbols  C  and  F  have  the  quality  that  they  indicate,  in  a  way, 

the  process  (Art.  22)  by  which  the  derivative  is  obtained.      The  symbol  — 

dx 
before  a  function  indicates  that  the  operation  of  differentiation  with  respect 
to  X  is  to  be  performed  on  the  function  ;  it  also  serves  to  indicate  the  result 
of  the  operation.  The  symbols  D  and  Dx,t  in  /and  J,  are  simply  abbrevia- 
tions for  the  symbol  — 
dx 

*  This  should  really  be  called  the  Hindoo  notation  ;  for  the  Arabs  obtained 
it  from  the  Hindoos.     See  Cajori,  Histortj  of  Mathematics. 

t  The  symbol  />^y  is  due  to  Louis  Arbogaste  (1750-1803),  professor  of 
mathematics  at  Strasburg.  The  symbol  -^  was  devised  by  Leibnitz,  and 
the  symbol  /',  by  Lagrange  (1736-1813),    ^* 


24.] 


BEPRt:SENTATION  OF  THE  DERIVATIVE. 


48 


Note  3.     Beginners  in  the  calculus  are  liable  to  be  misled  by  the  symbols 

2),  E,  G,  and  H,  especially  by  H.     The  symbol  -^  does  not  denote  a  fraction  ; 

dx 

it  does  not  mean  "the  ratio  of  a  quantity  dy  to  a  quantity  dx.''"'     Such  quan- 
tities are  not  in  existence  at  the  stage  when  -^  is  obtained.    It  should  be 

dx 

thoroughly  realized,  and  never  forgotten,  that   ^    is  short  for  —(y),    and 

dx  dx 

that  both  these  symbols  are  merely  abbreviations  for  lim_^j.^  —^  /gee  EaJf) 

Art.  22).     Some  one  has  remarked  that  the  dy  and  az  in  ^  are  merely  "  the 

dx 

ghosts  of  departed  quantities  ■' ;  but  perhaps  this  is  claiming  too  much  for 

them. 


24.  The  geometrical  meaning  and  representation  of  the  derivative 
of  a  function.  Let  f{x)  denote  a  function,  and  let  the  geometrical 
representation  of  the  function,  namely  the  curve 


be  drawn. 


y=f(^)y 


(1) 


Let  P(x^,  2/i)  and  Q(.Ti  +  Aa^i,  2/1  +  ^?/i)  be  two  points  on  the 
curve.     Draw  the  secant  LPQ.     Then 


tanPiX  = 


AXi 


Now  let  secant  LQ  revolve  about  P  until  Q  reaches  P.  Then 
the  secant  LP  takes  the  position  of  the  tangent  TP,  and  the 
angle  PLX  becomes  PTX ;  then,  also,  Aa\  reaches  zero. 


Hence 


tanPrX  =  lim 


Axi=0 


(2) 


44  INFINITESIMAL   CALCULUS.  [Ch.  111. 

Now  P(xi,yi)  is  any  point  on  the  curve;  hence,  on  letting 
(x,  y),  according  to  the  usual  custom,  denote  ariy  point  on  the 
curve,  and  ^  denote  the  angle  made  with  the  ic-axis  by  the 
tangent  at  (x,  y), 

tan<^  =  lim^,^o^-  (3) 

The  first  member  of  (3)  is  the  slope  of  the  tangent  at  any  point 
{x,  y)  on   the   curve   y=f(x),  and   the   second   member   is   the 

derivative  of  either  member  of  (1).     Hence  -^,  i.e.  /'(x),  is  the 

slope  of  the  tangent  at  any  point  (a?,  y)  on  the  curve  y  =/(x). 

This  principle  has  already  been  applied  in  the  exercises  in 
Art.  4. 

Curve  of  slopes.  If  the  graph  of  f'(x)  be  drawn,  that  is,  the 
curve  y=f'(x),  it  is  called  the  curve  of  sloj^es  of  the  curve 
y=if{x).  It  is  also  called  the  derived  curve,  and  sometimes  the 
differential  curve  of  y  =f{x).  For  instance,  the  curve  of  slopes 
of  the  curve  y  =  x^  is  the  line  y  =  2x.  The  curve  of  slopes  is 
the  geometrical  representative  of  the  derivative  of  the  function  ; 
the  measure  of  any  of  its  ordinates  is  the  same  as  the  slope  of 
y  =  f(x)  for  the  same  value  of  x. 

Ex.  Sketch  the  graphs  of  the  functions  in  Exs.,  Art.  22.  "Write  the 
equations  of  these  graphs.  Give  the  equations  of  their  curves  of  slopes,  and 
sketch  these  curves.    (Use  the  same  axes  for  a  curve  and  its  curve  of  slopes.) 

Note  1.  Produce  RQ  (Fig.  10)  to  meet  TP  in  8,  produce  PR  to  R  ,  and 
draw  R'Q'S'  parallel  to  RQ  to  meet  the  curve  in  Q'  and  TP  in  S'.     Then 

dx     /  ^  ^      PR      PR' 

Now,  if  Ax,  =  PR,^  =  ^;   and  if  Axi  =  PR',  ^  =  ^'    Also,      ' 
Aa^i      PR  '  Axi      PR' 

T  BQ     dy 

limp^^Op^  =  ^, 

,  ,.,       .  ,.  R'Q'     dy 

and  likewise,  hmp^j-^o  -p^  -  j-  • 

Note  2.  Hereafter,  in  general  investigations  like  the  above,  the  symbol  x 
will  be  used  instead  of  Xi  to  denote  any  particular  value  of  x  ;  and  similarly 
in  the  case  of  other  variables. 


25.]  MEANING   OF  THE  DERIVATIVE.  45 

25.  The  physical  meaning  of  the  derivative  of  a  function.  Sup- 
pose that  the  value  of  a  function,  say  s,  depends  upon  time ; 
i.e.  suppose  s=f(t). 

After  an  interval  of  time  A^,  the  function  receives  an  incre- 
ment As;  and  ,    .         ^,^  ,    .^. 


As  =f{t  +  M)  ~f(t). 

(1) 


.    A8^/(^  +  A0-/(0 
"   M  At 


lim^^^  fi.e.^)=f'(t).  (2) 


At  V        dt^ 

As 
Since  As  is  the  change  in  the  function  during  the  time  At,  — 

At 

is  the  average  rate  of  change  of  the  function  during  that  time. 
As  At  decreases,  the  average  rate  of  change  becomes  more  nearly 
equal  to  the  rate  of  change  at  the  time  t,  and  can  be  made  to 
differ  from  it  by  as  little  as  one  pleases,  merely  by  decreasing  A^. 
Hence  the  second  member  of  (2)  is  the  actual  rate  of  change 
at  the  time  t.  In  words :  The  derivative  of  a  function  with  respect 
to  the  time  is  the  rate  of  change  of  the  function. 

If  s  denotes  a  varying  distance  along  a  straight  line,  then  — 
denotes  the  rate  of  change  of  this  distance,  i.e.  a  velocity. 

(For  discussions  on  speed  and  velocity  see  text-books  on  Kine- 
matics and  Dynamics,  and  Mechanics.) 

Ex.     Show  that  if  s  =  Igt^,  then  —  =  gt.     (See  Art.  3  b.) 

dt 

Note.  Newton  called  the  calculus  the  Method  of  Fluxions.  Variable 
quantities  were  called  by  him  fluents  or  flowing  quantities,  and  the  rate  of 
flow,  i.e.  the  rate  of  increase  of  a  variable,  he  called  the  fluxion  of  the 

fluent.     Thus,  if  s  and  x  are  variable,  —  and  —  are  their  fluxions.    Newton 

dt  dt 

indicated  these  fluxions  thus :  5,  x.  This  notation  was  adopted  in  England 
and  held  complete  sway  there  until  early  in  the  last  century,  and  the  other 
notation,  that  of  Leibnitz,  prevailed  on  the  continent.  At  last  the  continental 
notation  was  accepted  in  England.  "The  British  began  to  deplore  the  very 
small  progress  that  science  was  making  in  England  as  compared  with  its 
racing  progress  on  the  continent.  In  1813  the  'Analytical  Society'  was 
formed  at  Cambridge.     This  was  a  small  club  established  by  George  Peacock, 


46  INFINITESIMAL   CALCULUS.  [Ch.  III. 

John  Herschel,  Charles  Babbage,  and  a  few  other  Cambridge  students,  to 
promote,  as  it  was  humorously  expressed,  the  principles  of  pure  '  D-ism,' 
that  is,  the  Leibnitzian  notation  in  the  calculus  against  tliose  of  'dot-age,' 
or  of  the  Newtonian  notation.     The  struggle  ended  in  the  introduction  into 

Cambridge  of  the  notation  ^,  to  the  exclusion  of  the  fluxional  notation  y. 

dx 
This  was  a  great  step  in  advance,  not  on  account  of  any  great  superiority  of 
the  Leibnitzian  over  the  Newtonian  notation,  but  because  the  adoption  of  the 
former  opened  up  to  English  students  the  vast  storehouses  of  continental 
discoveries.  Sir  William  Thomson,  Tait,  and  some  other  modern  writers 
find  it  frequently  convenient  to  use  both  notations."  —  Cajori,  History  of 
Mathematics,  page  283. 

26.   General  meaning  of  the  derivative :  the  derivative  is  a  rate. 

When  a  variable  changes,  a  function  of  the  variable  also  changes. 
A  comparison  .of  the  change  in  the  function  with  the  causal  change 
in  the  variable  will  determine  the  rate  of  change  of  the  function 
tvith  respect  to  the  variable.  The  limit  of  the  result  of  this  com- 
parison, as  the  change  in  the  variable  approaches  zero,  evidently 
gives  this  rate.  But  this  limit  has  been  defined  as  the  derivative 
of  the  function  with  respect  to  the  variable.  Accordingly  (see 
Art.  22,  Note  1),  the  main  object  of  the  differential  calculus  may  be 
said  to  be  the  determination  of  the  rate  of  change  of  the  function 
with  respect  to  its  argument. 

Note  1.  The  rate  of  change  of  the  function  with  respect  to  the  variable 
may  also  be  shown  in  a  manner  that  explicitly  involves  the  notion  of  time. 
In  the  case  of  the  function  y,  when  y  =/(x),  let  it  be  supposed  that  x  receives 
a  change  Ax  in  a  certain  finite  time  A^.  Accordingly  y  will  receive  a  change 
Ay  in  the  same  time  At.     Then,  from  the  equation  preceding  (e),  Art.  22, 

Ay  ^f(x  +  Ax)  -  fix)  ^  fix  4-  Ax)  -  f(x)     Ax 
At      'At  Ax  'At' 

When  At  approaches  zero.  Ax  also  approaches  zero.    On  letting  At  approach 

zero,  this  equation  becomes  (Art.  20,  Th.  c). 

dy 

dy     ^,,  ^dx     .     dir    dy    dx         ,^,  „^,  dii     dt 

-±=fi(x)—;  i.e.  ^  =  -:^  • —•       (1)  Whence,    /=^-       (2) 

dt     ''  ^  ''dt'         dt      dx    dt         ^  ^  '  dx     dx        ^  ^ 

dt 
Result  (1)  Can  also,  by  a  theorem  on  limits.  Art.  20  (d),  be  derived  from 

Ay  _Ay     Ax 
'Ax~~Ki^~Ai' 


26,  27.]  DIFFERENTIALS,  47 

Thus  the  derivative  of  a  function  with  respect  to  a  variable  may  be  regarded 
as  the  ratio  of  the  rate  of  change  of  the  function  to  the  rate  of  change  of  the 
variable. 

Note  2.  References  for  collateral  reading.  McMahon  and  Snyder, 
Diff.  C'a?.,  Arts.  88,  89;  Lamb,  Calculus,  Art.  33;  Gibson,  Calculus,  Arts. 
31^7,  51. 

EXAMPLES. 

1.  A  square  plate  of  metal  is  expanding  under  the  action  of  heat,  and 
its  side  is  increasing  at  a  uniform  rate  of  .01  inch  per  hour;  what  is  the 
rate  of  increase  of  the  area  of  the  plate  at  the  moment  when  the  side  is  16 
inches  long  ?     At  what  rate  is  the  area  increasing  10  hours  later  ? 

Let  X  denote  the  side  of  the  square  and  A  denote  its  area.     Then  A  =  x^. 

Now  M  =  M  .  Ax     ^^  dA^dA^  dx^      .    ^^  2a;  X  .01  sq.  inches 

At       Ax     At  dt       dx     dt  dt 

per  hour  =  .02  x  sq.  inches  per  hour.  Accordingly,  at  the  moment  when  the 
side  is  16  inches,  the  area  of  the  plate  is  increasing  at  the  rate  of  .32  sq.  inches 
per  hour.  Ten  hours  later  the  side  is  16. 1  inches ;  the  area  of  the  plate  is 
then  increasing  at  the  rate  of  .322  sq.  inches  per  hour.  The  area  of  the 
square  is  increasing  in  square  inches  2  x  times  as  fast  as  the  side  is  increasing 
in  linear  inches. 

2.  In  the  case  of  a  circular  plate  expanding  under  the  action  of  heat, 
the  area  is  increasing  at  any  instant  how  many  times  as  fast  as  the  radius  ? 
If  when  the  radius  is  8  inches  it  is  increasing  .03  inches  per  second,  at  what 
rate  is  the  area  increasing  ?  At  what  rate  is  the  area  increasing  when  the 
radius  is  15  inches  long  ? 

3.  The  area  of  an  equilateral  triangle  is  expanding  how  many  times  as 
fast  as  each  of  its  sides  ?  At  what  rate  is  the  area  increasing  when  each 
side  is  15  inches  long  and  increasing  at  the  rate  of  2  inches  a  second  ?  At 
what  rate  is  the  area  increasing  when  each  side  is  30  inches  long  and  increas- 
ing at  the  rate  of  2  inches  a  second  ? 

4.  The  volume  of  a  spherical  soap  bubble  is  increasing  how  many  times  as 
fast  as  its  radius  ?  At  what  rate  (cubic  inches  per  second)  is  the  volume  in- 
creasing when  the  radius  is  half  an  inch  and  increasing  at  the  rate  of  3  inches 
per  second  ?    At  what  rate  is  the  volume  increasing  when  the  radius  is  an  inch  ? 

5.  A  man  5  ft.  10  in.  high  walks  directly  away  from  an  electric  light  16 
feet  high  at  the  rate  of  3|  miles  per  hour.  How  fast  does  the  end  of  his 
shadow  move  along  the  pavement  ? 

27.   Differentials.     If  y=f(x),  (1) 

then,  in  accordance  with  notations  A  and  Hj  in  Art.  23, 

|=/(x).  (2) 


48  INFINITESIMAL    CALCULUS.  [Ch.  III. 

Suppose  that  an  arbitrary  difference  {i.e.  change  or  increment) 
h  may  be  made  in  the  independent  variable  x,  and  let  the  product 
f'ix)  •  h  be  denoted  by  k ;  that  is,  let 

lc=f{x)-h.  (3) 

(For  instance,  in  Fig.  10,  RS  =f'(x)  •  PE-,  here  h  =  PR,  and 
k  =  RS.  Also,  R'S'  =f{x)  .  PR' ;  here  li  =  PR',  and  k  =  R'S'). 
Now,  let  h  be  written  in  the  form  dx,  and  the  corresponding  value 
of  k  be  written  dy.     Then  (3)  is  written 

dp  =  f'{x)dx,  (4) 

This,  by  (2),  may  be  written 

dy  =  ^if.dx.  (5) 

dx  < 

As  used  in  Equation  (4),  dx  is  called  the  differeyitial  of  x,  dy  is 
called  the  differential  of  y,  and  f{x)dx  is  called  the  differential 
of  fix).  Since  f\x)  is  the  coefficient  of  dx  in  the  diiferential  of 
f{x),  fix)  is  frequently  called  the  differential  coefficient  of  f{x). 
(See  Art.  22.)  The  defining  Equations  (4)  and  (5)  may  be  ex- 
pressed in  words  :  Tlie  differential  of  a  function  y  of  an  indepen- 
dent variable  x  is  equal  to  the  derivative  of  the  function  multiplied 
by  the  differential  of  the  variable,  the  latter  differential  being  merely 
an  arbitrary  increment  (or  difference),  usually  small,  made  in  the 
variable. 

The  letter  d  is  used  as  the  symbol  for  a  differential ;  for  example, 
the  differential  of /(ic)  is  written  df(x) ;  thus  df(x)=f'(x)dx. 

Note  1.  It  is  highly  important  to  notice  that  in  Equations  (2)  and  (4), 
dy  and  dx  are  used  in  altogether  different  ways.*    In  (2)  and  (5),  -  ^  is  used 

as  a  symbol  for  liniAxio—  ;  and  it  denotes  the  definite  limiting  value  of  an 

Ax 

"indeterminate"  form  —    In  (4)  and  in  (5)  on  the  extreme  right  dx  is  not 

zero  (although  it  may  happen  to  be,  and  usually  is,  a  small  quantity),!  and 
the  dy  is  such  that  the  ratio  dy :  dx  is  equal  to  /'(x).    For  instance,  in  Fig.  10, 

*  In  one  respect  this  double  use  of  dx  and  dy  is  unfortunate  ;  for  it  tends 
to  confuse  beginners  in  calculus.     Other  notation  is  also  used. 

t  Later  on  many  examples  will  be  found  in  which  this  dx  is  an  infinitesimal. 


27.]  EXAMPLES.  49 

^^  of  Equation  (2)  is  tan  SPR.    As  to  Equations  (4),  (5),  if  dx  =  PR,  then  dy 

=  US',  and  if  dx=  PB',  then  dy  =  B'S'.  This  shows  that  dy,  in  (4),  is  the 
increment  of  the  ordinate  of  the  tangent  corresponding  to  an  increment  dx 
of  the  abscissa.  The  corresjjonding  increment  of  the  ordinate  of  the  curve 
y  —  f{x)  [i.e.  the  hicrement  of  the  function  /(x)]  in  some  cases  can  be 
found  exactly  by  means  of  the  equation  of  the  curve,  and  in  some  cases  can 
be  found,  in  general  only  approximately,  by  means  of  a  very  important 
theorem  in  the  calculus,  namely,  Taylor'' s  Theorem  (see  Chap.  XX.). 
Instances  of  the  former  are  given  below ;  instances  of  the  latter  are  given 
in  Art.  176. 

Note  2.  It  should  be  clearly  understood  that,  according  to  the  preceding 
remarks,  cancellation  of  the  dx^s  in  (5)  is  impossible. 

iV.B.  For  geometric  illustrations  of  derivatives  and  differentials  see 
Art.  67. 

EXAMPLES. 

1.  In  the  case  of  a  falling  body  s  =  I  gt^  (see  Art.  3)  ;  on  denoting,  as 
usual,  the  differential  of  the  time  by  dt,  ds,  the  corresponding  differential  of 
the  distance  is  [Ex.,  Art.  3  (6)]  gtdt ;  i.e.  ds  =  gldt.  The  actual  change  in  s 
corresponding  to  the  change  dt  in  the  time  is  [see  Eq.  (2),  Art.  3  (6)] 
gtdt  +  \g(idty\ 

2.  In  the  curve  y  =  x"^,  dy  =  2  x  dx.  The  actual  change  in  y  corresponding 
to  the  change  dx  in  x  is2xdx  -\-  (dxy.  (See  Eq.  (1),  Art.  4. )  Thus  if  a;  =  10 
and  dx  =  .001,  d^  =  2  x  10  x  .001  =  .02.  The  actual  change  in  the  ordinate  of 
the  curve  from  x  =  10  to  a;  =  10  +  .001  is  (10.001)2  ~  102,  j  g^  .020001.  This 
change  may  also  be  calculated  as  stated  above,  viz.  2  x  10  x  .001  +  (.001)2.  The 
dy  =  .02  is  the  change  in  the  ordinate  of  the  tangent  at  ic  =  10  from  x  =  10  to 
x  =  10.001  (see  Note  1).     (The  student  should  use  a  figure  with  this  example.) 

3.  Write  the  differentials  of  the  functions  in  the  Exs.  in  Art.  22. 

4.  Given  that  y  =  x^  —  4:  x^,  find  dy  when  x  =  4  and  dx  =  .1.  Then  find 
the  change  made  in  y  when  x  changes  from  4  to  4.1. 

5.  Given  that  y  =  2x^  +  7x^  —  9x  +  5,  find  dy  when  x  =  6  and  dx  =  .2. 
Then  find  the  change  made  in  y  when  x  changes  from  5  to  5.2. 

Note  3.  It  is  evident  from  these  examples  that  the  differential  of  a 
function  is  an  approximation  to  the  change  in  the  function  caused  by 
a  differential  change  in  the  variable  ;  and  that  the  smaller  the  differential 
of  the  variable,  the  closer  is  the  approximation.  When  the  differential  varies 
and  approaches  zero  it  becomes  an  infinitesimal. 

Ex.  Calculate  the  differentials  of  the  areas  in  Ex.  2,  Art.  26,  when  the 
differential  of  the  radius  is  .1  inch. 

Ex.  Calculate  the  differentials  of  the  areas  of  the  triangles  in  Ex.  3, 
Art.  26,  when  the  differential  of  the  side  is  .1  inch. 


50  INFINITESIMAL   CALCULUS.  [Ch.  III. 

Note  4.  It  may  be  remarked  here  that  in  problems  involving  the  use 
of  the  differential  calculus  derivatives  more  frequently  occur,  and  in  prob- 
lems in  integral  calculus  differentials  (viz.  infinitesimal  differentials)  are 
more  in  evidence. 

Note  5.    References  for  collateral  reading.    Gibson,  Calculus,  §  60 ; 

Lamb,  Calculus,  Arts.  57,  58. 

27  a.  Anti-derivatives  and  anti-differentials.  In  Arts.  22  and  27 
the  derivative  and  the  differential  of  a  function  have  been  defined, 
and  a  general  method  of  deducing  them  from  the  function  has 
been  described.  With  respect  to  the  derivative  and  the  differen- 
tial the  function  is  called  an  anti-derivative  and  an  anti-differential 
respectively.  Thus,  if  the  function  is  x^,  the  x-derivative  and  the 
i»-dift'erential  are  2  x  and  2  xdx  respectively ;  on  the  other  hand, 
Q(?  is  said  to  be  an  anti-derivative  of  2  a?  and  an  anti-differential  of 
2  xdx.  To  find  the  anti-derivatives  and  the  anti-differentials  of  a 
given  expression  is  one  of  the  two  main  problems  of  the  integral 
calculus.     (See  Art.  22,  Notes  1,  2,  and  Arts.  94,  96,  97.) 

Note.  Reference  for  collateral  reading.  Perry,  Calculus  for  Engi- 
neers, Arts.  12-24,  28,  30. 


CHAPTER   IV. 

DIFFERENTIATION    OF    THE    ORDINARY    FUNCTIONS. 

28.  In  this  chapter  the  derivatives  of  the  ordinary  functions  of 
elementary  mathematics  are  obtained  by  the  fundamental  and 
general  method  described  in  Art.  22.  Since  these  derivatives  are 
frequently  employed,  a  ready  knowledge  of  them  will  prevent 
stumbling  and  thus  make  the  subsequent  work  in  calcnlus  much 
simpler  and  easier;  just  as  a  ready  command  of  the  sums  and 
products  of  a  few  numbers  facilitates  arithmetical  work.  Accord- 
ingly these  derivatives  should  be  tabulated  by  the  student  and 
memorized. 

N.B,  The  beginner  is  earnestly  recommended  to  try  to  derive  these  results 
for  himself.    For  a  synopsis  of  the  chapter  see  Table  of  Contents. 

GENERAL  RESULTS  IN  DIFFERENTIATION. 

29.  The  derivative  of  the  sum  of  a  function  and  a  constant,  namely, 

Put  y  =  <fi{x)-\-c. 

Let  X  receive  an  increment  Ax;  consequently  y  receives  an 
increment,  ^y  say.     That  is, 

.-.  ^y^4>{x-\-^x)  +  c-  \_<i>{x)  4-  c] 

=  <^  (a;  +  ^x)  —  <f}(x). 

Ay  _  <f>(x  -{-  Ax)  —  <t>(x) 
' '  Ax  Ax 

51 


52  INFINITESIMAL    CALCULUS. 

Let  Ax  approach  zero  as  a  limit ;  then 


[Ch.  IV. 


lim^^^o 


Ay 

Ax 


<l>(x-\-  Ax)  —  <f>(x)^ 
Ax  ' 


I.e. 


I.e. 


I=*'(^)^ 


'^l<t>(.x)  +  c-]  =  4:X^(x)l 


(1) 


dx  dx 

Hence,  if  constant  terms  appear  in  a  function,  they  may  he  neg- 
lected when  the  function  is  differentiated. 

If  u  be  used  to  denote  <^  (a?),  result  (1)  can  be  expressed : 


doc  doc 


(2) 


CoR.  1.     It  follows  from  (1)  that  the  deriyative  of  a  constant  is 

zero.    This  may  also  be  derived  thus :    If  y  =  c  sl  constant,  then 

y-\-Ay  =  c',    and,   accordingly,    Ay  =  0.      Hence,   — ^  =  0   for   all 

d  d  ^^ 

values  of  Ax;  hence,  -^,  i.e.  —(c),  is  zero. 
dx  dx 

CoR.  2.  If  two  functions  differ  by  a  constant,  they  have  the 
same  derivative. 

From  (2)  and  Art.  27,  d(u  +  c)  =  du. 

Note  1,  In  geometry  y  =  c  is  the  equation  of  a  straight  line  parallel  to  the 
axis  of  X  and  at  a  distance  c  from  it.  The  slope  of  this  line  is  zero  ;  this  is  in 
accord  with  Cor.  1. 

Note  2.  The  curves  y  =  <p(x)  +  c,  in  which  c  is  an  arbitrary  constant 
(Art.  10),  can  be  obtained  by  moving  the  curve  y  =  ^{x)  in  a  direction 
parallel  to  the  ?/-axis.  The  result  (1)  shows  that  for  the  same  value  of  the 
abscissa,  the  slope  -^  is  the  same  for  all  the  curves.  See  Figs.  11,  12, 
below.  ^^ 


Fig.  11. 


Fig.  12. 


30.]  DIFFERENTIATION   OF  FUNCTIONS.  53 

Note  3.     The  converse  of  Cor.  1  is  also  true ;  namely,  if  the  derivative  of 
a  quantity  is  zero,  the  quantity  is  a  constant. 
Ex.    Show  this  geometrically.     (See  Art.  24.) 

Note  4.  The  converse  of  Cor.  2  is  also  true ;  namely,  if  two  functions 
have  the  same  derivative,  the  functions  differ  only  by  an  arbitrary  constant. 
(By  the  same  derivative  is  meant  the  same  expression  in  the  variable  and  the 
fixed  constants.)     For  let  <f>(x)  and  F{x)  denote  the  functions,  and  put 

y  =  <p{^)-  F{x). 

By  hypothesis,  Dy  =  tp'{x)  -  F'{x)  =  0. 

Hence,  by  Note  3,  y  =  c  ; 

and  accordingly,  <f>{x)  =  F(x^  +  c. 

Ex.    Show  this  geometrically. 

Note  5.  If  ^  =  <f>'Mi  then  y  =  <p(x)  +  c,  in  which  c  denotes  any  con- 
dx 
stant.  Hence  0(a;)  +  c  is  a  general  expression  for  all  the  functions  whose 
derivatives  are  <^'(x).  Functions  such  as  0(x)  +  1,  ^(x)  —  3,  obtained  by 
giving  particular  values  to  c,  are  particular  functions  having  the  same  deriva- 
tive 0'(x). 

Note  6.  Notes  4  and  5  come  to  this  :  The  anti-derivatiye  of  a  function 
is  indefinite,  so  far  as  an  arbitrary  additive  constant  is  concerned. 

30.   The  derivative  of  the  product  of  a  constant  and  a  function,  say 

c<|>(a5). 

Put  y  =  c<f>(x). 

Let  X  receive  an  increment  Ax;  consequently  y  receives  an 
increment,  Ay  say. 

That  is,  y -{- Ay  =  c<f>{x -[-  Ax). 

.'.Ay  =  cl<f>(x-^Ax)-<f,(x)2. 


I.e. 


.  Ay ^    rA(x-\-Ax)-cf>(x)l 
"  Ax        [_  Ax  J 

ii,n         ^.'^-lim         J<t>(x-^Ax)-<f> 

iim^j^o-T—  —  ^1"1ax=o  ^    7 

Ax  [_  Ax 

dx 


M]; 


i.e.  ^[c^{x)-\  =  e4>\x).  (1) 


54  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

That  is,  the  derivative  of  the  product  of  a  constant  and  a  function 
is  the  iwoduct  of  the  constant  and  the  derivative  of  the  function. 

If  <^  (x)  be  denoted  by  u,  then  (1)  is  written 

In  particular,  \i  u  =  x,  —  (ex)  =  c. 
ax 

From   the   above   and   the   definition   in  Art.  27,  d[c<l>(x)^  = 
cd[<^(aj)],  d(cu)  =  cdu,  d(cx)  =  cdx. 

Ex.     See  Exs.,  Art.  22. 


31.   The  derivative  of  the  sum  of  a  finite  number  of  functions,  say 

Put  y=^(^x)-^F(x)  +  ->: 

Then,  on  giving  x  an  increment  Ax  (as  in  Arts.  29,  30), 

y -\- Ay  =  ct>(x -\-  Ax)  +  F(x  +  Ax)  +  -.  -. 

.-.  Ay  =  (t>(x  +  Ax)  —  ct)(x)  +  F(x  +  Ax)  —  F(x)  -\ . 

.  Ay  ^  cf>(x  4-  Ax)  -  <t>(x)      F(x  +  Ax)  -  F(x)       ^_^ 
Ax  Ax  Ax 

Hence,  on  letting  Ax  approach  zero, 

dx     dx  dx 

i.e.  £l^(x)  +  F(x)  +  ...:\=.l>'(x)+F'(x)  +  ....  (2) 

That  is,  the  derivative  of  a  sum  of  a  finite  number  of  functions 
is  the  sum  of  their  derivatives. 

If  the  functions  be  denoted  by  u,  v,  iv,  •••,  i.e.  if 

y  =  u-^v-{-tv-\ , 

the  result  (1)  may  be  expressed  thus : 

dp  _  du    dv  .  dw  ■ 
dx     dx    dx     dx 


31,32.]  DIFFERENTIATION  OF  FUNCTIONS.  55 

rrom  this  and  Art.  27, 

dy  =  du  +  dv  +  dtv  +  ••• . 

Note  1.  The  differentiation  of  the  sum  of  an  infinite  number  of  functions 
is  discussed  in  Art.  173. 

In  working  the  following  exercise  the  result  of  Ex.  8,  xVrt.  22,  may  be 
used. 

Ex.   Find  the  derivatives  of 

2  x3  +  7  x2  -  10  a;  +  11,  x^  -  17  x  +  10,  _  a;2  +  21  x  -  5. 

32.   The  derivative  of  the  product  of  two  functions,  say  ^{x)F{x). 

Put  y  =  <i>(x)F(x). 

Then,  on  giving  x  an  increment  Ax, 

y  +  Ay  =  <i>{x  -f  Ax)F{x  +  A:^). 

.-.  Ay  =  4>(x  4-  Ax)F(x  -\-  Ax)  -  <l>(x)F(x). 

.  Ay  ^  ct>(x  +  Ax)F(x  +  Ax)  —  <f>(x)F(x)  .^. 

"  Ax''  Ax  '  ^  ^ 

On  letting  A.r  =  0,  the  second  member  takes  the  form  -•     In 

order  to  evaluate  this  form,  introduce  (f>(x  -h  Ax)F(x)  —  <fi{x  -\- 
Ax)F(x)  in  the  numerator  of  this  member.*  Then,  on  combining 
and  arranging  terms,  (1)  becomes 

Ax     ^^  ^  Ax  ^  ^  Ax 

Hence,  on  letting  Ax  approach  zero, 

f^=  .i>{x)F'(x)  +  F(x),l>Xx).  (2) 

That  is :  The  derivative  of  the  product  of  two  functions  is  equal  to 
the  product  of  the  first  by  the  derivative  of  the  second  plus  the 
product  of  the  second  by  the  derivative  of  the  first. 

*  Equally  icell,  ^(x)  F{x  +  Ax)  —  <t>{x)  F(x  +  Ax)  may  be  thus  introduced. 
The  student  should  do  this  as  an  exercise. 


5Q  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

If  the  functions  be  denoted  by  u  and  v,  that  is,  if 
y  =  uv, 
then  (2)  may  be  expressed 

^  =  u^  +  v^.  (3) 

di€         doc        doc  ^ 

The  derivative  of  the  product  of  any  finite  number  of  functions 
can  be  obtained  by  an  extension  of  (3).     For  example,  if 

y  z=  nvw, 
then,  on  regarding  vw  as  a  single  function, 

-^  =  (vw) \-  u —  (vw) 

dx  dx        dx 


du  , 

vw f- 

dx 


(    dv  ,     div\ 

U[  lU—-{-  V ) 

y    dx        dxj 


du  ,        dv  .       dw  /.. 

=  vw h  i^u h  uv — •  (4) 

dx  dx  dx 

Similarly,  \i   y  =  uvwz, 

dy  du  ,  dv  ,         dw  ,  dz  /e-x 

-^  =  vwz 1-  uwz h  uvz h  uvw —  (5) 

dx  dx  dx  dx  dx 

In  general :  In  order  to  find  the  derivative  of  a  product  of  several 
functions,  multiply  the  derivative  of  each  function  in  turn  by  all 
the  other  functions,  and  add  the  results. 

Note.     Another  way  of  obtaining  (5)  is  given  in  Art.  39  (a). 

The  differential  of  the  product  of  two  functions.     If 

y  =  uv, 
then,  from  (3)  and  the  definition  in  Art.  27,  it  follows  that 

dy  =  u'^dx-^v—dx,  (6) 

dx  dx 

But,  by  Art.  27,         —dx  =  dv,  and  —dx  =  du. 
dx  dx 

Hence,  (6)  may  be  written 

d(uv)  =  udv  +  vdu,  (7) 


33.]  DIFFERENTIATION  OF  FUNCTIONS,  57 

Similarly,  if  y  =  uvw, 

it  follows  from  (4)  that     dy  =  vwdu  -f  wudv  -j-  uvdw. 

On  division  by  uvw,  this  takes  the  form 

d{uvw)      du  .  dv  ,  dw  ,q. 

= 1 1 {O) 

uvtv         uvw 

Ex.  1.    Write  dy  in  forms  (7)  and  (8),  when  y  =  uvwz. 

Ex.  2.  Differentiate  (x^  +  l){x^  —  2  a:  +  7)  by  the  above  method  ;  then 
expand  this  product  and  differentiate,  and  show  that  the  results  are  the 
same. 

Ex.  3.   Treat  the  following  functions  as  indicated  in  Ex.  2  : 

x\x  -1)(7^  +  4),  (aa;2  -\-bx  +  c){lx  +  m). 

Ex.  4.   Write  the  differentials  of  the  functions  in  Exs.  2,  3. 

33.  The  derivative  of  the  quotient  of  two  functions,  say  <|>(a?)  -^  Fix), 

Put  y=ii^. 

Then,  on  proceeding  as  in  Arts.  29-32, 

"     ^      F{x-^Ax)      F(x) 

^  <^(a;  4-  Ax)F(x)  -  <f,(x)F(x  4-  Aa;)^ 
~  F{x)F(x  +  Ax) 

.  Ay  ^  <i>{x  +  Ax)F(x)  -  <f>(x)F(x  +  Ax)  ^. 

*  *  Ax  F{x)F{x  +  Aa:)Aic  '  ^ 

On  letting  Ax  =  0,  the  second  member  takes  the  form  -•  In 
order  to  evaluate  this  form,  introduce 

F(x)<t>(x)  -  F(x)<f>(x) 

in  the  numerator  of   this  member.      Then,  on   combining   and 
arranging  terms,  (1)  becomes 

^(^T^(..  +  Ax)  -  <A(x-)n_  ^(^Tj'Cx  +  Ax)  -  F(x)1 
Ax~  F(x)F{x  +  Ax) 


68  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

Hence,  on  letting  Ax  approach  zero, 

dy ^F(x)<f>Xx)  -  <f>(x)F'(x)  ,.v 

dx  lF(x)y  ^  ^ 

That  is :  If  one  function  be  divided  by  another,  then  the  derivative 
of  the  fraction  thus  for^ned  is  equal  to  the  product  of  the  denomi- 
nator by  the  derivative  of  the  numerator  minus  the  product  of  the 
numerator  by  the  derivative  of  the  denominator,  all  divided  by 
the  square  of  the  denominator. 

If  the  functions  be  denoted  by  u  and  v ;  that  is,  if    * 
then  (1)  has  the  form 


u 


dy  _    doc        ddc  (2) 

dic~  v2 

The  differential  of  the  quotient  of  two  functions.     If  2/  =  ->  t^^^i 


from  (2)  and  the  definition  in  Art.  27, 


V 


V  —  dx  —  u — dx  /Q\ 

dx  dx  yy) 

^y  = ^2- — 

But,  by  Art.  27,  —dx  =  du  and  ^^dx  =  dv.     Hence  (3)  may 
,  ...  dx  dx 

be  written 

ay^vdu-udv^  (4) 

Note.     The  derivative  (1),  or  (2),  can  also  be  obtained  by  means  of  Art. 

32.     For  if    y  =  -,   then  vi/ =  u.      Whence  v^^  +  y~  =  —-      From  this 
V  dx        dx     dx 

dy  ^ldu_ydv^  which  reduces  to  the  form  in  (2)  on  substituting  -  for  y. 
dx     V  dx      V  dx  V 

Ex.  1.    Find  the  derivatives  and  the  differentials  of 

a;^  3;2  +  7  x  -U 

3a:2-7a;  +  2'   jc'^  +  8'    2x;^-9x  +  S 

Ex.  2.    Calculate  the  differentials  of  the  functions  in  Ex.  1  when  x  =  2 

and  dx  =  .1. 


34.]  DIFFERENTIATION  OF  FUNCTIONS.  69 

34.   The  derivative  cf  a  function  of  a  function. 

Suppose  that  y  =  <l>(ti), 

and  that  u  =  F{x), 

and  that  the  derivative  of  y  with  respect  to  x  is  required.  (Here 
<^(m)  and  F(x)  are  continuous  functions.)  The  method  which 
naturally  comes  first  to  mind,  is  to  substitute  F(x)  for  u  in  the 
first  equation,  thus  getting  y  =  <f)[F(x)'],  and  then  to  proceed 
according  to  preceding  articles.  This  method,  however,  is  often 
more  tedious  and  difficult  than  the  one  now  to  be  shown. 

Let  X  receive  an  increment  Ax ;  accordingly,  n  receives  an  incre- 
ment Aw,  and  y  receives  an  increment  Ay.     Then 

y-\-Ay  =  <t>(u  +  Au). 

.'.  Ay  =  (f>(;u  +  Au)  -  <l>(u). 

Ay  _<t>(u  H-  All)  —  <f>(u) 
' '  Ax~  Ax 

<l>(u  -\-  Au)  —  <f>(u)     Aw 
Au  Ax 

Kow  Ax,  Au,  Ay  reach  the  limit  zero  together.  Hence  (Art.  20, 
Th.  c)  on  letting  Ax  approach  zero, 

dx     du^^^  '-^  dx 

i,e.^  =  ^.^.  (1) 

dor.     du    dx  ^  ^ 

Note.  It  should  be  clearly  understood  that  the  first  member  of  (1)  does 
not  come,  and  cannot  come,  from  the  second  member  by  cancellation  of  the 
dw's.     Cancellation  is  not  involved  at  all. 

Result  (1),  which  may  be  expressed  more  emphatically  (Art.  23), 

is  an  important  one  and  has  frequent  applications.  It  may  be  thus  stated  : 
the  derivative  of  a  function  icith  respect  to  a  variable  is  equal  to  the  product 
of  the  derivative  of  the  function  with  respect  to  a  second  function  and  the 
derivative  of  the  second  function  with  respect  to  the  first  named  variable. 
(Here  all  the  functions  concerned  are  supposed  to  be  continuous.) 


60  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

From  (1)  and  (2)  it  results  that 

A  (2,)  ^ 

due  dx 

Relations  (1)  and  (2),  Note  1,  Art.  26,  are  special  applications  of  (1)  [or 
(2)  and  (3)].    The  showing  of  this  is  left  as  an  exercise  for  the  student. 

Ex.  1.    Explain  why  the  dw's  in  (1)  may  not  be  cancelled. 

Ex.  2.    Find   ^,  given  that  y  =  u^  and  ii  =  x^  -i- 1. 
dx 

Here  ^^  =  Sn%  —=2x.     . •.  ^  =  6  u^x  =  6 x^x"^  +  1)2. 
du  dx  dx 

Ex.  3.    Find   ^  when  y  =  S  iC^  and  u  =  x^-ox-{-l.  .  Verify  the  result 
dx 
by  the  substitution  method  referred  to  at  the  beginning  of  the  article. 

Ex.  4.    Find    —    when  z  =  2v'^  -Sv  +  1   and  v  =  (jt^-\-  1.     A^erify  the 

result  by  the  substitution  method. 

Ex.  5.    Show  that  a  function  of  a  function  is  represented  by  a  curve  in 
space.    (See  Echols,  Calculus,  Appendix,  Note  2.) 

35.   The  derivative  of  one  variable  with  respect  to  another  when 
both  are  functions  of  a  third  variable. 

Let  X  =  Fit)  and  y  =  <f>(t). 

Now  — ^  =  -^  -; Now  A^,  Ax,  and  Ay  reach  the  limit  zero 

Ax     At      M 

together. 

Hence,  Art.  20,  Th.  d,  on  letting  At  approach  zero, 

dy 

dy^dt^  (1) 

dx     dx 
dt 

This  result  may  also  be  derived  as  a  special  case  of  result  (3), 
Art.  34.     This  is  left  as  ah  exercise  for  the  student. 

Ex.  1.   Find  -^  when  y  z=St'^  -  7  t -\- 1,  2ind  x  =  2t^  -  ISt^  +  lit. 
Here^  =  6«-7,  ^  =  6«2_26«+ll.     .-.  ^  = ^-^^ 


dt  dt  .  dx     6f-^-26«+ll 

Ex.  2.    Find  ^  when  x  =  2t^  +  \lt-\  and  j/  =  3  «*  -  8  «2  _(-  9. 
dx 

Ex.  3.    Find  —  when  m  =  7x*  -  3  and  v  =  3x2  +  14x  -  4. 
dv 


35-37.]  DIFFERENTIATION  OF  FUNCTIONS.  61 

36.  Differentiation  of  inverse  functions.  If  y  is  a  function  of  x, 
then  X  is  a  function  of  y;  the  second  function  is  said  to  be  the 
inverse  function  of  the  first.  This  is  expressed  by  the  following 
notation:  If  y=f(x),  then  x=f~^{y).  Examples  of  inverse 
notation  have  been  met  in  trigonometry. 

The  equation  _J^  .  — ^  zzz  1  is  always  true.      Accordingly  (Art. 

Ax     Ay 

20,  Th.  c),  ^  .^'  =  1. 
'  ^'   dx    dy 


Hence, 


dx     doc 
dy 


DIFFERENTIATION   OF   PARTICULAR  FUNCTIONS. 

In  the  following  articles  u  denotes  a  continnous  fanction  of  or, 

and  differentiation  is  made  with  respect  to  x.     The  letters  a,  n,  •••, 
may  denote  constants. 

N.B.     It  is  advisable  for  the  student  to  try  to  obtain  the  derivatives  before 
having  recourse  to  the  book  for  help. 

# 

A.   Algebraic  Functions. 
37.   Differentiation  of  ii^, 
(a)    For  71,  a  positive  integer. 
Put  2/  =  ^*" ; 

i.e.  y  =  uuu  •••  to  w  factors. 

...  ^  =  u^-^  —  +  u^-^  ^  -f-  . . .  to  n  terms      (Art.  32) 
du  dx  dx 

„^idu 
dx 

In  particular,     —  (x)  =  1,  and  —  (x")  =  na;**~^. 
dx  dx 

Ex.  1.    Give  the  derivatives  with  respect  to  x  of 

w2,     3  w*,     7  i<9,     x8,     3  xS     7  a;i2,     9  x^  -  17  x'^  +  10  x  +  40. 


62  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

Ex.  2.    Find  the  a;-derivative  of  (2x  +  ly^. 

On  denoting  this  function  by  !/,  and  putting  ii  iov  'Ix  +  1,  y  =  u^^.    Hence 

dx  dx 

Now  ^'  =  2  ;   hence      ^  =  36  w"  =  36  (2  x  +  7)i7. 
dx  dx 

The  substitution  w  for  2  a;  +  7  need  not  be  explicitly  made.     For,  if 

y=(2x+7)i8, 

then  ^  =  18  (2  x  +  7)i'  ^^  (2x4-7)     (Art.  34) 

dx  dx 

=r  36  (2  X  +  7)1". 
Ex.  3.    Differentiate 

(5  x2  -  10)2-1,     (-3  ^4  4.  2)1^     (4  x2  +  5)^(3  x^  -  2  x  +  7)-\ 
(6)    i^or  n,  a  negative  integer.     Let  n  =  —  m,  and  put  ?/  =  u"^. 
Then 

; — (Art.  33) 


y  = 

u 

-.=1. 

dii 

ir 

•t(1)- 

dx 

dx 

u- 

— 

u^-         ■ 

= 

nu 

=(-»)«<— I 


Ex.  4.    Differentiate  with  respect  to  x, 
w-2,     w-7,     M-11,     x-7,     3x-5,     17a:-l^     (^2  -  3)-*,     (3x*  +  7)-5, 

3x5_7x3  +  2-UV^3' 
X      X''      9x"* 

(c)    Fo7'  n,  a  rational  fraction.     Let  n  =  --,  in  which  p  and  q 
are  integers. 

Put  y=n^;  then  ?/« =  ?/,^. 

On  differentiating,      qy'^-^-^  ^^pu^-^—  • 

dy  _  1)  u^~^  da  _  p   u^-'^    du  _  p  g~^du  _      n-idu 
"  dx      q  y^'^  dx     q     ^(g-i)  dx      q        dx  dx 


37.]  DIFFERENTIATION   OF  FUNCTIONS.  63 

Ex.  5.    Find  the  ^-derivatives  of 

Vw  (i.e.  zt2),     u~^,     M^,     Vx,     x^,     Vx^,     VSx'^  —  d, 

i/2x^  +  7 X  -  3,     V2 a;  +  7,     (3x-7)~5,     3:^2- 7x2  +  A  +  A  _  ..2_. 

x^      a:^      7x^ 

(d)    For  n,  an  incommensurable  number.     In  this  case  it  is  also 

true  that  — (w«)  =  nu''-'—.     This  is  proved  in  Art.  39  (6). 
dx  dx  ^ 

Hence,  for  all  constant  values  of  n, 

£(u")  =  nun-.2.  (1) 

In  particular,  if  u  =  x,  —  (a.-")  =  wa;'*"^ 
ax 

Ex.  6.    Find  the  x-derivatives  of 

M^2,    X^^    5  X^^    (2  X  +  5)^5,    (3  x2  +  7  X  -  4)^3. 

Ex.  7.    Write  three  functions  which  have  x^  for  a  derivative. 
Ex.  8.    Do  as  in  Ex.  7  for  the  functions 

x5,  1,  VS,  ^/x%  ^x,6x^---  i-. 

'  x2         '         '        '  x-^      Vx 

Ex.  9.  Show  that  the  general  form  lohich  includes  all  the  functions  that 
have  x'^  for  the  derivative^  is 1-  c,  in  lohich  c  is  an  arbitrary  constant. 

71  +  1 

Note  1.  The  result  (1)  and  the  general  results,  Arts.  29-36,  suffice  for 
the  differentiation  of  any  algebraic  function. 

Note  2.  Case  (a)  can  also  be  treated  as  follows :  Put  y  =  m",  and  let  x 
receive  an  increment  Ax ;  then  u  and  y  receive  increments  An  and  Ay 
respectively.     Then  y  -^  Ay  =  (u  +  Aw)'*-     On  expanding  the  second  member 

by  the  binomial  theorem,  then  calculating  Ay  and  then  -^,  and  finally  letting 
Ax  approach  zero,  the  result  will  be  obtained. 

Note  3.     It  is  well  to  remember  that  —  (x)  =  1   and  —  (Vx)  =— — 

dx  dx  2  Vx 

Ex.  10.   Do  the  operations  indicated  in  Note  2. 


Ex.  11.    Differentiate  "^    •     Find  the  value  of  the  derivative  when 

ic  =  2.  Vx"-^  +  2 

Put  y^x(x^  +  7)^. 

(x2  +  2)^ 


64  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

(a;2  +  2)^-^[x(a;2  +  7)^]  -  a;(x2  +  7)^--  (x^  +  2)^ 
rj^hen  ^  = <^^  ^^ 

On  performing  the  differentiations  indicated  in  the  second  member,  and 
reducing,  it  is  found  that 

dy_     4  a;*  +  19  g2  -f.  42 

^^      3  (a;2  +  7)*(a;2  4-  2)^ 

Hence,  when        a;  =  2, 

-^=:1.68,  approximately. 
dx 

Ex.  12.    Differentiate  the  following  functions  with  respect  to  x : 

(2x-5)(x2+llx-3),  ax'«+-^,  ^-^t£!,  ^_z:^,   VIT^,  -^+5^-7x5, 

ic"    1  —  ^2    a  4-  ic  x* 

^^  +  ''',  — =^=,  — ^,  a/t"^^'  (1  +  ^^'^)"'  («  +  &^^)S  x-(l  -  x)«, 

(a  +  a:)  Va  —  cc. 

Ex.  13.    Find  ^  when  x^^^^  +  2a;  +  3y  =  5.    Here  y  is  a?i  implicit  function 
dx 

of  X.     On  differentiation  of  both  members  with  respect  to  x, 

a^2#  (y^)  +  y^^  (X^)   +  2  4-  3^  ::=:  0  ; 

c?x  dx  dx 

i.e.  3  X V  ^  ^_  2  xy3  +  2  +  3  ^  =  0. 

dx  dx 

dy         2(l  +  xi/3) 
From  this  - ^ — — ^-^• 


dx         3  (1  +  x^y'^) 


dy 


Ex.  14.   (a)  Find  ^  when  x  and  y  are  connected  by  the  following  rela- 
dx 
tions :    ?/3  +  x3  -  3  ax«/  =  0  ;  X*  +  2  ax'^y  -ay^  =  0;  7  x'^y^  +  2  x?/3  _  3  x^y  +  4  x^ 
-82/2  =  5;  (a  +  yy^b'^  -  y^)  +  (x  +  a)2?/2  =  0  ;  x2  +  2/2  =  a2  ;  a2?/2  +  fe'^a^^  ^ 

a2&2.     In  the  last  case  also  obtain  -^  directly  in  terms  of  x. 

dx 

(6)  In  the  ellipse  3  x2  +  4  y2  =  7,  find  the  slope  at  the  points  (1,  1), 
(1,  -1),  (-1,1),  (-1,  -1). 

N.B.  The  following  examples  should  all  be  worked  by  the  beginner. 
They  will  serve  to  test  and  strengthen  his  grasp  of  the  fundamental  prin- 
ciples of  the  subject,  and  will  give  him  exercise  in  making  practical  applica- 
tions of  his  knowledge.     For  those  who  may  not  succeed  in  solving  them 


37.]  DIFFERENTIATION  OF  FUNCTIONS.  65 

after  a  good  endeavour,  two  examples  are  worked  in  the  note  at  the  end  of 
the  set. 

Ex.  15.  A  ladder  24  feet  long  is  leaning  against  a  vertical  wall.  The  foot 
of  the  ladder  is  moved  away  from  the  wall,  along  the  horizontal  surface  of 
the  ground  and  in  a  direction  at  right  angles  to  the  wall,  at  a  uniform  rate 
of  1  foot  per  second.  Find  the  rate  at  which  the  top  of  the  ladder  is  descend- 
ing on  the  wall  when  the  foot  is  12  feet  from  the  wall. 

Ex.  16.  Show  that  when  the  top  of  the  ladder  is  1  foot  from  the  ground, 
the  top  is  moving  575  times  as  fast  as  when  the  foot  of  the  ladder  is  1  foot 
from  the  wall. 

Ex.  17.  Find  a  curve  whose  slope  at  any  point  (x,  y)  is  2x.  Find  a 
general  equation  that  will  include  the  equations  of  all  such  curves.  Find 
the  particular  curve  which  passes  through  the  point  (1,  2). 

Ex.  18.  A  man  standing  on  a  wharf  is  drawing  in  the  painter  of  a  boat  at 
the  rate  of  4  feet  a  second.  If  his  hands  are  6  feet  above  the  bow  of  the  boat, 
how  fast  is  the  boat  moving  when  it  is  8  feet  from  the  wharf  ? 

Ex.  19.  A  man  6  feet  high  walks  away  at  the  rate  of  4  miles  an  hour  from 
a  lamp  post  10  feet  high.  At  what  rate  is  the  end  of  his  shadow  increasing 
its  distance  from  the  post  ?     At  what  rate  is  his  shadow  lengthening  ? 

Ex.  20.  A  tangent  to  the  parabola  y'^  =  16  x  intersects  the  x-axis  at  45°. 
Find  the  point  of  contact. 

Ex,  21.  A  ship  is  75  miles  due  east  of  a  second  ship.  The  first  sails  west 
at  the  rate  of  9  miles  an  hour,  the  second  south  at  the  rate  of  12  miles  an 
hour.  How  long  will  they  continue  to  approach  each  other  ?  What  is  the 
nearest  distance  they  can  get  to  each  other  ? 

Ex.  22.  A  vessel  is  anchored  in  10  fathoms  of  water,  and  the  cable  passes 
over  a  sheave  in  the  bowsprit  which  is  12  feet  above  the  water.  If  the  cable 
is  hauled  in  at  the  rate  of  a  foot  a  second,  how  fast  is  the  vessel  moving 
through  the  water  when  there  are  20  fathoms  of  cable  out  ? 

Ex.  23.  Sketch  the  curves  y^  =  4:X  and  x^  =  4 1/,  and  find  the  angles  at 
which  they  intersect.  (If  6  denotes  the  angle  between  lines  whose  slopes 
are  m  and  n,  tan^  =(m  —  w)-^(l  +  mn)  ;  see  analytic  geometry  and  plane 
trigonometry.) 

Ex.  24,   Sketch   the   curves  y^  =  Sx  and  x^  =  8 y, 
and  find  the  angles  at  which  they  intersect. 

Note.    Examples  worked.    Ex.  15.    Let  FT  be 

the  ladder  in  one  of  the  positions  which  it  takes  during 
the  motion,  and  let  FH  be  the  horizontal  projection  of 
FT.     Let  FH=x,  and  HT=y.     Then 

x2  -h  2/2  =  576.  (1)  Fig.  13. 


QQ  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

Now  X  and  y  are  varying  with  the  time  ;  the  time-rate  —  is  given,  and 

dv  ^^ 

the  time-rate  -^  is  required.     Differentiation  of  both  members  of  (1)  with 

respect  to  the  time  give 

dt  dt 

whence  dy^_xdx^  ^  ,^. 

dt         y  dt 
In  this  case,  —  =c  1  foot  per  second,  x=  \2  feet,  and,  accordingly, 

Cit 


y  =  V242  -  12-^  feet  =  12  V3  feet. 

.*.  -^  = •  1  foot  per  second  =  —  .577  feet  per  second. 

d^  12  V3 

The  negative  sign  indicates  that  y  decreases  as  x  increases.     It  should  be 
noticed  that  the  result  (2)  is  general,  and  that  all  particular  solutions  can 

be  derived  from  it  by  substituting  in  it  the  particular  val^ies  of  x,  y,  and  — • 

dt 
Ex.  17.   Find  a  curve  whose  slope  at  any  point  (x,  y)  is  2x.     Find  a 
general  equation  that  will  include  the  equations  of  all  such  curves ;  and  find 
the  particular  curve  which  passes  through  the  point  (1,  2). 

Here  ^=2x. 

dx 

Hence  y  =  x^  +  c,  **  (1) 

in  which  c  denotes  any  arbitrary  constant.  This  is  the  general  equation  of 
all  the  curves  having  the  slope  2  x.  .-.  y  =  x^  +  T  is  one  of  the  curves ; 
y  =  x'^  —  b  is  another.  If  the  point  (1,  2)  is  on  one  of  the  curves  (1),  then 
2  =  1+0;  whence  c  =  1,  and,  accordingly,  y  =x^  -{-  1  is  the  particular  curve 
passing  through  (1,  2).  As  in  Ex.  15  it  is  easier  to  find  first  the  general  solu- 
tion of  the  problem  in  question,  and  therefrom  to  obtain  any  particular 
solution  that  may  be  required.    Figure  12  shows  some  of  these  curves. 

B.    Logarithmic  and  Exponential  Functions. 

38.    Note.     To  find  lim/w=ao  f  1  +  —  J  .     This  limit  is  required  in  what 
follows.  ^        '"^ 

(a)  For  m,  a  positive  integer.     By  the  binomial  theorem, 

\        mj  m  1.2         m^  1.2.3  w»»  ^  ^ 

This  can  be  put  in  tlie  form 

l(l_n      ifi_lVi_2\ 

fl+l\"'=l4.1  +  _A «l/  +  J WV ral  ^2) 

\       ml  2  1  3  1  ^  ^ 


38,  39.]  DIFFERENTIATION  OF  FUNCTIONS.  67 

On  letting  m  approach  infinity,  and  taking  the  limits,  this  becomes  * 

lim„^«  (l  +!)'"=  1  +  1  +±  +  ±+  ... 
\        mj  2  !      3  ! 

=  2.718281829....  (3) 

This  constant  number  is  always  denoted  by  the  symbol  e. 

(&)  The  result  (3)  is  true  for  all  infinitely  great  numbers,  positive  and 
negative,  integral,  fractional,  and  incommensurable.  For  the  proof  of  (3) 
for  all  kinds  of  numbers,  see  Chrystal,  Algebra  (ed.  1889),  Part  II.,  Chap. 
XXV.,  §  13,  Chap.  XX VIII.,  §§  1-3  ;  McMahon  and  Snyder,  Diff.  Gal., 
Art.  30,  and  Appendix,  Note  B ;  Gibson,  Calculus^  §  48. 

Note  on  e.  The  transcendental  number  e  frequently  presents  itself  in 
investigations  in  algebra  (for  instance,  as  the  base  of  the  natural  logarithms, 
and  in  the  theory  of  probability),  in  geometry,  and  in  mechanics.  The  num- 
bers e  and  it  are  perhaps  the  two  most  important  numbers  in  mathematics. 
They  are  closely  allied,  being  connected  by  the  very  remarkable  relation 
gtn-  —  —  l,t  which  was  discovered  by  Euler.  See  references  above,  and  Klein, 
Famous  Problems  (referred  to  in  footnote.  Art.  8),  pages  55-67. 

39.   Differentiation  of  \oga  u. 

Put  y  =  log.w, 

and  let  x  receive  an  increment  ^x ;  then  u  and  y  consequently 
receive  increments  A?*  and  A^/  respectively. 

Then        y  -\-  ^y  =  log„  (ii  -\-  Au). 

,'.  A?/  =  log„  (u  -\-  Au)  -  log«  u 


=  log.(^)=log.(l  +  ^). 


Ax  \         u  J    Ax 

1     u 

On  introducinor Au  in  the  second  member. 


u    Alt 

u 

Ay  _  1      71   ■,        /-.      A? A     A?fc  _  1  1   ^    A      Ai*\  A^    Au 
Ax      a    Alt       "  v         u  I    Ax      it     *"  V         u  I       Ax 


*  This  conclusion  is  properly  reached  only  after  a  more  rigorous  investiga- 
tion than  is  here  attempted.     (See  Arts.  167-171.) 
t  See  Art.  179. 


68  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

From  this,  on  letting  Aic  approach  zero  and  remembering  that  Aw 
and  A?/  approach  zero  with  I^x,  it  follows  by  Arts.  22,  23,  38,  that 

dx     u         "      dx^ 

liu  =  x,  then  -^  (loga  a?)  =  — .  loga  e. 

Ifa  =  e,  then  ^(log«^)  =  ^^. 

If  7^ = ic,  and  a = e,  then  -^(log  a?)  -  — . 

Note.     When  e  is  the  base  it  is  usual  not  to  indicate  it  in  writing  the 
logarithm. 

Ex.  1.    Find  the  derivatives  of  log„  (3  x^  +  4  x  -  7),  log  (3  a;^  +  4  x  -  7), 
logio  (3  ^2  +  4  X  —  7).     Find  the  values  of  these  derivatives  when  x  =  3. 


Ex.  2.     Find  the  values  of  the  derivatives  of  log  Vx^  +  10,  logio  vx^  +  10, 
when  X  =  2. 

Ex.  3.     Differentiate  the  following:    log^-^l^,    logJi-i-^,    log^-t^, 
log  (x  4-  y/yi^  +  «■'),  log  (log  x),  X  log  X.  ^  "^  ^  1-x  i_Vx 

Ex.  4.    Find  anti-derivatives  of       ^^^  '"^ — ,       '^  ^^  ~  '^    ,  — . 

x2  +  3x  +  5    x3-7x-l    2x 

(a)    Logarithmic  differentiation.     If 

?/  =  uvw,  (1) 

then  log  y  =  log  ?«  +  log  v  +  log  w. 

On  differentiation,     l^  =  l*f  +  l*'  +  l*^, 
2/dic     i^da;      vdx     wdx 


whence  ^=  „^„ri*'  +  l*'+ l*^']. 

da;  \_udx     vdx     wdxj 


(2) 


This  result  can  easily  be  reduced  to  the  form  obtained  in 
Art.  32.  The  same  method  can  be  used  in  the  case  of  any  finite 
number  of  factors.     This  method  of  obtaining  result  (2)  is  called 


40.]  DIFFERENTIATION   OF  FUNCTIONS,  69 

the  method  of  logarithmic  differentiation.  It  is  frequently  more 
expeditious  than  that  given  in  Arts.  32,  33,  especially  when 
several  factors  are  involved. 

Ex.  5.    Find  ^  when        y  =  ^i(Ei±I)_.  (See  Ex.  11,  Art.  37.) 

Here,  log  y  =  log  x  +  |  log  (x^  +  7)  -  ^  log  (x2  +  2). 

On  differentiation,  1^  =  1  +  ^ 2x 

ydx     X     x2  +  7      3  (x2  +  2) 

From  this,  on  transposing,  combining,  and  reducing, 

dy  ^     4  x^  +  19  a;2  4-  42 

^     3  (x2  +  7)^(x2  +  2)^ 
Ex.  6.    Differentiate,  with  respect  to  x,  the  following  functions : 


(«)    t^+2)! ;     (6)   {x-l){x-2)  V2x  +  5^7x-5, 

(6)   Differentiation  of  an  incommensurable  (constant)  power  of  a 
function.     This  paragraph  is  supplementary  to  Art.  37  (d). 

Let  y  =  u^^ 

in  which  n  is  any  constant,  commensurable  or  incommensurable. 
Then  logy  =  n  log  ?«. 


From  this 

Idy^ 
ydx 

ndu^ 
'  udx' 

and  hence 

doc 

_  ny  du  _ 
u  dx 

-nu^- 

^du^ 
dx 

40.   Differentiation  of 

a«. 

Put 

y  = 

=  a". 

Then 

log  2/  = 

=  u  log  i 

%. 

On  differentiation, 

Idy^ 
ydx 

:loga  . 

du 
dx 

.  dy  _ 

"dx 

:2/loga.^^ 
dx 

le. 

ax 

ax 

70  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

If  u=:iX,  then  —  {a^)  =  a^  •  log  a, 

due 

If  a  =  e,  then  -^  (et*)  =  e**— • 

doc  dx 

If  i<  =  aj,  and  a  =  e,  then 

that  is,  the  derivative  of  e""  is  itself  e^. 

Note  1.  On  the  derivation  of  results  in  Arts.  39,  40.  The  derivative 
of  loga  u  was  deduced  by  the  general  and  fundamental  method,  and  has 
been  used  in  finding  the  derivative  of  a".  The  latter  derivative  can  be 
found,  however,  by  the  fundamental  method,  independently  of  the  deriva- 
tive of  loga  u.  Moreover,  the  derivative  of  loga  u  can  be  obtained  by  means 
of  the  derivative  of  a"-.  These  various  methods  of  finding  the  derivative 
of  a'^  and  log^  u  are  all  employed  by  writers  on  the  calculus.  For  examples 
see  Todhunter,  Diff.  Gal.,  Arts.  49,  50;  Gibson,  Calculus,  §65,  where  both 
these  derivatives  are  obtained  independently  of  each  other ;  Williamson, 
Diff.  Cal.,  Arts.  29,  30;  McMahon  and  Snyder,  Diff.  Cal,  Arts.  30,  31, 
where  the  derivative  of  the  logarithmic  function  is  first  obtained  and  the 
derivative  of  the  exponential  function  is  deduced  therefrom  ;  and  Lamb, 
Calculus,  Arts.  35  (Ex.  5),  42,  where  the  derivative  of  the  exponential 
function  is  obtained  first  and  the  derivative  of  the  logarithmic  function 
is  deduced  therefrom.     (See  also  Echols,  Calculus,  Art.  33  and  foot-note.) 

Note  2.  On  the  expansion  of  e^  in  a  series  see  Hall  and  Knight,  Higher 
Algebra,  Art.  220;  Chrystal,  Algebra,  Vol.  II.,  Chap.  XXVIII.,  §§4,  5;  and 
other  texts.     (This  expansion  is  derived  by  the  calculus  in  Art.  178,  Ex.  7.) 

Ex.  Assuming  the  expansion  for  e^,  show  that  the  derivative  of  e^  is 
itself  e^. 

Note  3.  The  compound  interest  law.  The  function  e'  "is  the  only 
[mathematical]  function  known  to  us  whose  rate  of  increase  is  proportional 
to  itself  ;  but  there  are  a  great  many  phenomena  in  nature  which  have  this 
property.  Lord  Kelvin's  way  of  putting  it  is  that  '  they  follow  the  compound 
interest  law.'  "  (See  Hall  and  Knight,  Higher  Algebra,  Art.  234,  and,  in 
particular,  Perry,  Calculus,  Art.  97  and  Art.  98,  Exs.  4,  2.) 

Ex.  1.   Differentiate,  with  respect  to  x,  e^"",  \(f,  10'^''',  e^*. 

Ex.  2.    Find  the  ^-derivatives  of  e^,  lO'',  e''+^  10^+^ 

Ex.3.    Find  the  a;-derivatives  of  the  following: 

C'x^,   a*",  — ^— ,   a;e-^,   £llL.£l!,   £!!. 
e*  —  1  e""  -\-  e-*     x 

Ex.  4.   Find  anti-derivatives  of  e^^,  a;e*^  2  e^+\ 


41,  42.]  DIFFERENTIATION  OF  FUNCTIONS.  71 

41.   Differentiation  of  u^,  in  which  u  and  v  are  both  functions 
of  X, 

Put  y  =  u\  (1) 

Then  log  y  =  v  log  u. 

On  differentiation,    1  ^  =  ^  ?^  +  log  «  .  *!. 
?/  da;      u  dx  dx 


(ia;        \it  da; 


cix  +  '°°"-£" 


Note  1.  It  is  better  not  to  memorize  result  (2),  but  merely  to  note  the 
fact  that  the  function  in  (1)  is  easily  treated  by  the  method  of  logarithmic 
differentiation. 

Note  2.     The  beginner  needs  to  guard  against  confusing  the  derivatives 

of  the  functions  m",  a'*,  and  m«. 

Ex.  1.    Find  -^  when  y  =  oC'. 

clx 

Here  log  y  =  x  log  x. 

1  dy     X 
On  differentiation,  -  -^  =  -  -i-  log  x  ; 

y  dx     X        "     ' 

whence  -^  =  a;*  (1  +  log  x) . 

Ex.  2.   Find  the  ^-derivatives  of 

(3x  +  7r^    (3x+7)^   {(3x  +  7)*}2,    ^x,   x-\   e'',    f^^^   log^. 

\x/  a* 

(7.   Trigonometric  Functions. 
42.   Differentiation  of  sinu. 
Put  2/  =  sin  u. 

Then  y  +  Ay  =  sin  (w  +  Aw). 

.-.  Ay  =  sin  (w  +  Au)  —  sin  w 

=  2  cos  [  1*  H-  -— )  sin  — —     (Trigonometry) 


72  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

.*.  -^-  =  2  COS    1^  +  — —    sm  — -  •  — 
Aa;  V         2  /  2      Aa; 


cos 


.    Aw 
sm— - 

2      ^u 


Au        Ax 


2 
Let  Aa;  =  0 ;    then  also  Au  =  0,  and 


sm 


Aw 


Hm^^^ -^  =  lim^^^o cos  ( i^ -|- -~]  •  lim^„^o  — r •  Hm^^^o t^*  ; 

Ax  V  2  /  Aw  Aa; 


I.e.  -^  =  cos  u  '  1  *  —  : 

dx  dx 


2 


i.e.  ^(sin«e)  =  cos«*^.  (1) 

dx  dx  ^ 

In  particular,  if  w  =  a;, 

^(siii£c)  =  cosa;.  (2) 

dx  ^ 

That  is,  the  rate  of  change  of  the  sine  of  an  angle  with  respect 
to  the  angle  is  equal  to  the  cosine  of  the  angle. 

Note  1.  Result  (2)  can  also  be  obtained  by  geometry.  (Ex.  Show  this.) 
See  Williamson,  Dif.  Cal,  Art.  28,  and  other  texts. 

Note  2.     Result  (2)  shows  that  as  the  angle  x  increases  from  0  to  —  the 

rate  of  increase  of  the  sine  is  positive,  since  cos  x  is  then  positive.     As  x 

increases  from  ^  to  tt  the  rate  is  negative  (i.e.  the  sine  decreases),  since 

2  fj^ 

cos  X  is  then  negative.     The  rate  is  negative  when  x  increases  from  w  to  '-|— , 

q  2 

and  the  rate  is  positive  when  x  increases  from  —  to  2  t.     This  agrees  with 

what  is  shown  in  elementary  trigonometry,  and  it  is  also  apparent  on  a 
glance  at  the  curve  y  =  sin  x. 

Note  3.  Result  (2)  also  shows  that  if  the  angle  increases  at  a  uniform 
rate,  the  sine  increases  the  faster  the  nearer  the  angle  is  to  zero,  and 
increases  more  slowly  as  the  angle  approaches  90°.  This  is  also  apparent 
from  an  inspection  of  a  table  of  natural  sines,  or  from  a  glance  at  the  curve 
y  =  sin  X. 

Note  4.  The  derivative  of  sin  u  has  been  found  by  the  general  and 
fundamental  method  of  differentiation.     It  is  not  necessary  to  use  this 


Ex,  4.    Find  the  x-derivatives  of  — — -,  x  sin  2  x. 


43.]  DIFFERENTIATION  OF  FUNCTIONS.  73 

method  in  finding  the  derivatives  of  the  remaining  trigonometric  and  anti- 
trigonometric  f unctions, .  for  these  derivatives  can  be  deduced  from  that  of 
the  sine. 

Ex.  1.    Find  the  x-derivative^of  sin  2  u,  sin  3  u,  sin  i  u,  sin  |  w,  sin  y  u. 
Ex.  2.    Find  the  cc-derivatives  of  sin2x,  sin3aj,  sin^a;,  sin  3x2,  sin2  3x, 
sin4x^  sin5  4x. 

Ex.  3.    Find  the  derivatives  with  respect  to  t  of  sin  5 1,  sin  1 1^. 

x2sinfx  +  -V 
sin3x  \        4/ 

Ex.  5.    At  what  angles  does  the  curve  y  =  sinx  cross  the  x-axis  ? 

Ex.  6.  At  what  points  on  the  curve  y  =  sinx  is  the  tangent  inclined  30° 
to  the  X-axis. 

Ex.  7.  Draw  the  curve  ?/  =  sin  2  x.  At  what  angles  does  it  cross  the 
X-axis  ? 

Ex.  8.  Draw  the  curve  y  =  sinx  -\-  cos  x.  Where  does  it  cross  the  x-axis  ? 
At  what  angles  does  it  cross  the  x-axis  ?     Where  is  it  parallel  to  the  x-axis? 

Ex.9.  Find  the  x-derivatives  of  the  following:  sin  nx,  sinx",  sin»»x, 
sin(l  4- ic2),  sin(wx  +  a),  sin(a+6x''),  sin=^4x,  ^^^,  sin(logx),  log(sinx), 
sin(e^)  •  logx.  ^ 

Ex.  10.    (a)  Find  anti-derivatives  of 

cosx;  cos3x,  cos(2x-K5),  xcos(x2  — 1). 

(5)  Find  anti-differentials  of  cos2xdx,  cos(3x  — 7)dx,  x-cosx^^x. 

Ex.  11.  Calculate  dCsinx)  when  x  =  46°  and  dx  =  20',  and  compare  the 
result  with  sin  46°  20' -  sin  46°.  (Radian  measure  must  be  used  in  the 
computation.) 

Ex.  12.    Compare  d{sin  x)  when  x  =  20°  and  dx  =  30',  with 

sin  20°  30'  -  sin  20°. 

43.   Differentiation  of  cosu. 

Put  y  =z  cos  u. 


Then 


2/  =  sm(| 


l=^^<i-^^)l(i-^)   [^^t-^^^^^-w] 


-Sint.^; 


I.e. 


^(co»ti)^^$inu^-  (1) 

ax  dx  ^  ^ 


74  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

In  particular,  if  u  =  x, 

-^  (cos  a?)  =  - sin  05.  (2) 

Ex.  1.    Obtain  derivative  (1)  by  the  fundamental  method. 

Ex.  2.  Show  that  result  (2)  agrees  in  a  general  way  with  what  is  shown 
in  trigonometry  about  the  behaviour  of  the  cosine  as  the  angle  changes  from 
0°  to  360''.    Also  inspect  the  curve  y  =  cos  x. 

Ex.  3.  Find  where  the  curve  y  =  cos  x  is  parallel  to  the  x-axis,  and  where 
its  slope  is  tan  25°. 

Ex.  4.  Show  that  the  tangents  of  the  curve  y  =  cos  x  cannot  cross  the 
X-axis  at  an  angle  between  4-  45°  and  +  135°. 

Ex.  5.  Find  the  slope  of  the  tangent  to  the  ellipse  x  =  a  cos  0,  y  =  h  sin  d. 
(See  Art.  35.) 

Ex,  6.  Find  the  slope  of  the  tangent  to  the  cycloid  x  —  a(d  —  sind)^ 
1/  =.a(l  —  cos  5).     What  angle  does  this  tangent  make  with  the  x-axis  when 

a  =  5,  and  6  =-? 
3 

Ex.7.   Find  the  x  derivatives  of  the  following:    cos(2x -f  5),  cos^5x, 

x^cosx,      ~  ^^^  ^,  cosmxcosnx,  xe'^^^',  e^'cosmx. 
1  +  cosx 

Ex.8.  Find  anti-differentials  of  sinxdx,  sin^xdx,  sin  (3x  —  2)(?x, 
X  sin  (x2  -1-  i)dx. 

Ex.  9.  Calculate  d  cos  x  when  x  =  57°  and  dx  =  30',  and  compare  the 
result  with  cos  57°  30'  -  cos  57°. 

44.  Differentiation  of  tan  u. 

Put  y  =  tan  u. 

Then  y  =  5il^. 

cosu 

COS  u — (sin  u)  —  sin  u  —  (cos  ii) 
dy^___dx___ dx^         ^ 

dx  cos^u 

_  (cos'^  1^4-  sin^  u)  du 


=  sec^  u  -— ; 


I.e. 


cos^  u  dx  dx 

-^(tanw)=sec2t*^.  (1) 

dx  dx 


44-4C.]  DIFFERENTIATION  OF  FUNCTIONS,  75 

Uu=x,  then  ^ (tan a?)  =  sec^ x.  (2) 

Ex.  1.  Show  the  agreement  of  result  (2)  with  the  facts  of  elementary 
trigonometry,  and  with  the  curve  y  =  tan  x. 

Ex.  2.  Show  that  the  tangents  of  the  curve  y  =  tan  x  cross  the  x-axis  at 
angles  varying  from  +  45°  to  +  90°. 

Ex.  8.  State  the  x-derivatives  of  tan  2  w,  tan  3  u,  tan  mu,  tan  nii-,  tan  2  x, 
tan  ^  x,  tan  wix,  tan  3  x^,  tan  4  x^,  tan  wix**,  tan^  3  x,  tan^  4  x,  tan"  mx, 
tau2(|x  +  3),  log  tan-. 

Ex.  4.   Find  anti-differentials  of  sec2xdx,  sec2  2  x  dx,  sec^  (3  x  +  a)(?x. 

Ex.  5.^  Compute  d  tan  x  when  x  =  20°,  dx  =  20',  and  compare  the  result 
with  tan  20°  20'  -  tan  20°. 

Ex.  6.    When  is  the  differential  of  tan  x  infinitely  great  ? 

45.  Differentiation  of  cot  w. 

Either,  substitute  ^^^  ^\  for  cot  w,  and  proceed  as  in  Art.  44 ; 
sin  u 

or,  substitute  tan  (90° — u)  for  cot  w,  and  proceed  as  in  Art.  43 ; 

or,  substitute  for  cot  u,  and  differentiate.     It  will  be 

foupdthat  ^^^'' 

4-(icoiu)  =  -cosec^u^*  (1) 

dx  dx  ^  ^ 

If  uz=x,  -^  (cot  x)  =  -  cosec^  x,  (2) 

dx  ^  ^ 

Ex.  Show  the  general  agreement  of  result  (2)  with  the  facts  of  ele- 
mentary trigonometry,  and  with  the  curve  y  =  cot  x. 

46.  Differentiation  of  sec  u. 

Put    _  y  =  sec  u  = 

cosu 


Then 


dy  _  sin  u    du  _     1       sin  u    du  , 
dx     co^^u    dx     cosw    cosi^    dx^ 


i.e.  ^(secM)  =  secwtaiiw^.  (1) 


dx  dx 

dx 


liu  =  x,  -^  (sec  a?)  =  see  ^  tan  a?.  (2) 


T6  INFINITESIMAL   CALCULUS,  [Ch.  IV. 

47.   Differentiation  of  esc  u. 

Put  2/=csc«=^.     Then^  =  -''5^*i. 

siiiw  cZa;         ^m^udx 

That  is,  —  (esc  u)  =  -  esc  «t  cot  u—-  (1) 

dx  dx  ^  ^ 

If  II  =  X,  — (esc  a;)=:— esc  x  cot  x.  (2) 

dx  ^  ■ 


Note.     Or  put  y  =  esc  u  =  sec  (  ^  —  «  | ,  and  proceed  as  in  Art. 


43. 


48.  Differentiation  of  vers  u.     Put  y  =  vers  tc=:l  —  cos  u.     Then, 
on  differentiation,  .  . 

—  (vers  u)  =  sin  u  — • 
dx  dx 

In  particular,  ii  u  =  x, 

— -  (vers  oc)  =  sin  a?. 
doc 

Ex.  1.    Find  the  ic-derivatives  of  cot  (2  x  +  3),  sec  (|  x  +  3),  esc  (3  cc  -  7), 
vers  (5  X  +  2),  sec»*x. 

Ex.  2.    Find  the  ^derivatives  of  cot2  (3^  +  1),  sec^  (i  ^  _  1),  esc-'  |(«  +  5), 
cot  (9^2),  sec  (7  «  -  2)2. 

Ex.  3.    Show  that  D  log  (tan  x  +  secx)=  D  log  tan  (i  tt  +  |  x)  =  sec  x. 

D.    Inverse  Trigonometric  Functions.* 

49.  Differentiation  of  sin"^w. 

Put  2/  =  sin~^i^. 

Then  sin  y  =  u. 

On  differentiation,      cos  2/^  =  —. 

dx     dx 

'    ^.V_     1     du  _  1  du^ 

dx     cosy  dx      v'l  —  sin^y  ^^' 

I.e.  :^  (sin-i  t*)  =       ^       ^^.  (1) 

Ifi^  =  a;,  ^rsin-icc)=        ^       .  (2) 

*  See  Murray,  Plane  Trigonometry,  Arts.  17,  88. 


47-50.] 


DIFFERENTIATION   OF  FUNCTIONS. 


77 


Note  1.  On  the  ambiguity  of  the  derivative  of 
sill  -la?.  The  result  in  (2)  is  ambiguous,  since  the  sign  of 
the  radical  may  be  positive  or  negative.  This  ambiguity 
is  apparent  on  looking  at  the  curve  y  =  sin-i  x,  Fig.  14. 

Draw  the  ordinate  ABODE  at  x  =  Xi.  The  tangents 
at  B  and  D  make  acute  angles  with  the  oj-axis,  and  the 
tangents  at  C  and  E  make  obtuse  angles  with  the  x-axis. 

Hence,  at  B  and  D  ^  is  positive  ;  and  at  C  and  E  -^  is 
dx  dx 

negative.     That  is,  at  B  and  D  —  (sin-i  x)  =  —  "^         ; 

dx  Vl 


and  at  C  and  E  —  (sin-i  x)  = 

dx  vl 

d 


Thus  the  sign 


x{' 


of  —  (sin-i  x)  depends  upon  the  particular  value  taken  of  the  infinite  number 
of  values  of  y  which  satisfy  the  equation  y  =  sin-i  x. 

Note  2.  If  it  is  understood  that  there  be  taken  the  least  positive  value  of 
y  satisfying  the  equation  y  =  sin-i  Xi  (in  which  Xi  is  positive),  then  the  sign 
of  the  derivative  is  positive.     Similar  considerations  are  necessary  m  (1). 


Ex.  1.    Show  by  the  graph  in  Fig.  14,  or  otherwise,  that  when  x  =  1, 


dx 


(sin-i  x)  =-\-  CO,  and  that  when  x 
Ex.  2.    Find  the  a;-derivatives  of 

.-1^  +  1     «i 


1,  —  (sin-i  x)  is  —  Qo. 
dx 


sm~^  aj",    sm- 


V2 


2x 

1+^2' 


sm^ 


2x 


VT 


sin-i  vl  —  x'^,    Vl  —  x'^  •  sin-i x  —  x,  sin-^  vsin x. 

Ex.  3.    Show  that  a  tangent  to  the  curve  y  =  sin-^  x  cannot  cross  the 
ic-axis  at  an  angle  between  —  45°  and  +  45°. 

'^2 


Ex,  4.   Find  anti-derivatives  of 


'J.X 


vT 


c2    y/\-x^    y/l-y^ 


50.   Differentiation  of  cos~^(/. 

Put  y  =  cos"^  u. 

Then  cos  y  =  u. 

On  differentiation,   —  sin  ?/  —  =  — • 

dx     dx 


dy  _  _    1     du 
dx         sin  y  dx 


du 


Vl  —  cos^  y  ^^ 


78  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

i.e.  4-  (cos-i  «*)  = — ^• 

ax  y/\  _  1^2  dx 

li  u  =  x,  —  (cos-i)  =  -^       - 


dx  VI  -  x^ 

Ex.  1.  Explain  the  ambiguity  of  sign  in  the  derivative  of  cos"i  x  by 
means  of  the  curve  y  =  cos~i  x.  Show  that  if  there  be  taken  the  least 
positive  value  of  y  satisfying  y  =  cos"i  x,  in  which  x  is  positive,  the  sign 
of  the  derivative  is  negative. 

Ex.  2.   Determine  the  angles  at  which  the  tangents  touching  the  curve 

y  =  cos"i  X  where  x  —  — ,  cross  the  a:-axis. 

V2 

/v.2n  1  1    3»2  n  a* 

Ex.  3.   Find  the  ^-derivatives  of  cos"i ,  cos-i ,  a  cos~i . 

x2«  + 1'  1  +  a;2'  a 


Put  y  =  tan~^ 

Then  tan  y  =  u. 

On  differentiation,        sec^  y—  =  — 

dx      dx 


,  dy  _     1     du  _         1         du  ^ 

dx      sec^  ydx     1  +  tan^  y  dx ' 

i.e.  /(tan-i|*)  =  -i— ^. 

dx  1  +  «*2  dx 

In  particular,  if  w  =  x, 

4-  (tan-i  X)  =  :r-^' 

dx  1  +  x^ 

Note.    The  derivative  of  tan-i  x  is  always  positive.    This  is  also  evident 
on  a  glance  at  the  curve  y  =  tan"i  x. 

Ex.  1.   Find  the  a;-derivatives  of  tan-i2a;,  tan-i  2  y,  tan-^x^,  tan-i  y^. 

Ex.  2.   Find  the  ^derivatives  of  tan-i  4  t,  tan"i  <*,  tan~i  3  x^. 

Ex.  3.   Show  that  the  angles  made  with  the  ic-axis  by  the  tangents  to 
the  curve  y  =  tan-i  x  are  0°,  45°,  and  the  angles  between  0°  and  45°. 

Ex.  4.    Show  how  to  determine  the  abscissas  of  the  points  oi  y  =  tan-i  x, 
the  tangents  at  which  cross  the  x-axis  at  an  angle  of  30°. 

2x  ~ 

Ex.  6.    Find  the  ^-derivatives  of  the  following  :  tan~i -,  tan-i 


1  _  a;2'  1  +  a;2 


tan-i-^=,  tan-i^l+^^-\  ,,,-iJ_^,  tan-i  4^^?^. 


51-53.]  DIFFERENTIATION   OF  FUNCTIONS.  79 


Ex.6,    (a)  Show  that  2)tan-iA/^^ cos^^l       (ft)  Show,  by  differenti- 
al +  cos  x     2 

ation,  that  D  ( tan-i  x  +  tan-i  -  J  is  independent  of  x. 
Ex.  7.    Find  anti-differentials  of     ^        2xdx      x^  dx 


l+x2'    1+x*'    l  +  a:8 

52.  Differentiation  of  cot~^  u.  On  proceeding  in  a  manner  simi- 
lar to  that  in  Art.  51,  it  will  be  found  that 

^(cot-ite)  =  --i-f^. 
doc  1  +  «*2  dop 

Ex.  1.  Show,  by  means  of  the  curve  y  =  cotix,  that  the  derivative  of 
cot-i  X  is  always  negative. 

Ex.  2.   Find  the  x-derivative  of  cot-i  -  +  log  \' 

53.  Differentiation  of  sec  ^  u. 

Put  y  =  sec"^  u. 

Then  sec  y  =  u. 

dv     du 
On  differentiation,  sec  y  tan  y  --  =  —-. 

(tX       (XX 

^    dy  1  du  _  1  du 

"  dx~ sec y  tan y  dx~ gee y  Vsec^ y-ldx' 

i.e.  ^(sec-ii*)  = ^=^'  (1) 

If  u  =  x,  then  ^  (sec-i  x)  =       1        «  (2) 

Ex.  1.  Explain  the  ambiguity  of  the  result  (2).  Show  that,  when  x  is 
positive,  the  positive  value  of  the  radical  is  taken  with  the  least  positive 
value  of  sec-i  x. 

Ex.  2.   Find  the  x-derivatives  of  sec-^  x^,   sec-i ,   sec-i        ^ 


sec-i-^  ^    . 
x-2-1 

1 


Ex.  3.    Show    by    differentiation    that    tan 


independent  of  x.  ^1  -  ^^  ^1  -  ^^ 


80  INFINITESIMAL   CALCULUS.  [Ch.  IV. 

54.  Differentiation  of  cosec"^  u.  On  proceeding  in  a  manner 
similar  to  that  in  Art.  53,  it  will  be  found  that 

^(csc-i«*)  =  -— J^ ^.  (1) 

doc  u  Vw2_i  dx  ^  ^ 

If  u  =  x,  -^(csc-iiZ5)  = 1__.  (2) 

dx  ^V^23T  ^  ^ 

Ex.  1.  Explain  the  ambiguity  in  sign  in  (2)  by  means  of  the  graph  of 
csc-i  u.  Show  that,  when  x  is  positive,  the  negative  value  of  the  radical  is 
taken  with  the  least  positive  value  of  csc-i  u. 

55.  Differentiation  of  vers~^  u. 


t.e, 


Put 

y  =  vers-^  u. 

Then 

vers  y  —  u. 

On  differentiation, 

sin3  =  f^. 
dx     dx 

.    dy_ 

X     du              1           du 

dx 

sm  y  ax      VI  _  cos^  y  dx 
1                  du 

-VI  -  (1  -  yeis  yy  dx- 

^ 

rvers-ii^)-         1         ^^. 

(1) 

dx 

V2u-  «*2  dx 

li  u-x,              ^ 

(jers-'^x)  -         -^ 

(2) 

'             dx 

V2x-x^ 

Ex.  1.   Find  the  x-derivative  of  vers-i   ^^^  - 

l+x2 

56.   Differentiation  of  implicit  functions  :  two  variables. 

N.B.  Examples  of  the  differentiation  of  implicit  functions  have  been 
given  in  Exs.  13,  14,  Art.  37.  A  preliminary  study  of  these  examples  will 
help  to  make  this  article  clear. 

Let  y  be  an  implicit  function  of  x,  the  function  y  and  the 
variable  x  being  connected  by  a  relation 

fix,y)  =  c.  (1) 


54-56.]  DIFFERENTIATION  OF  FUNCTIONS.  81 

If,  as  sometimes  happens,  it  is  impossible  or  inconvenient  to 
express  y  as  an  explicit  function  of  x,  the  derivative  -^  may  be 
obtained  in  the  following  way  : 

On  taking  the  ar-derivative  of  each  member  of  (1),  there  is 
obtained  a  result  of  the  form 

(2) 


P+Qf  = 

dx 

0. 

rom  this 

dy_ 
dx 

:- 

P 

Q 

Since  the 

a;-derivative  of  f{x,  y) 

is 

p 

(3) 

Q—,  the  differential  of 

f(x,  y)  is  (Art.  27)  Pdx-\-Q^dx,  i.e.  (Art.  27)  Pdx+Qdy. 

dx 

Ex.  1.     Find  -^,  when  xy  =  c. 

Differentiation  of  the  members  of  this  equation  gives  y  +  x-^=:0;  whence 

-^  =  — ^.     The  x-derivative  of  xy  is  y -{- x~  :  accordingly,  the  differential 

dx         X  dx 

of  xy  is  xdy  +  ydx.     [Compare  result  (7),  Art.  32.] 

Ex.  2.  Write  the  differentials  of  the  first  members  of  the  equations  in 
Exs.  13,  14,  Art.  37. 

Ex.  3.     Find  y^    in   each   of  the  following  cases  :     (i)    x^  +  y^ 
(ii)  x^  -\-yi  =  J;     (iii)  |^  + 1^  =  1  J     (iv)   (cos  xy  -  (sin  yy  =  0. 

Ex.  4.  Write  the  differentials  of  the  first  members  of  the  equations  in 
Ex.  3. 

Note  1.  It  should  be  observed,  as  illustrated  in  Equation  (2)  and  the 
above  examples,  that  when  the  differential  of  /(x,  y)  is  written  Pdx  +  Qdy, 
P  is  the  same  expression  as  is  obtained  by  differentiating /(x,  y)  with  respect 
to  X,  and  at  the  same  time  regarding  y  as  constant  or  letting  y  remain 
constant,  and  Q  is  the  same  expression  as  is  obtained  by  differentiating 
/(x,  y)  with  respect  to  y,  and  at  the  same  time  regarding  x  as  constant  or 
letting  X  remain  constant.  Here  P  is  called  the  partial  x-derivative  of  /(x,  y), 
and  Q  is  called  the  partial  y-derivative  of  /(x,  y).     These  partial  derivatives 

are  denoted  by  the  symbols    -^  ^^'  ^^  and    -^  ^^'  ^^  respectively.     With  this 

dx  dy 

notation,  result  (3)  may  be  written 


(4) 


df(x,  y) 

or 

wJ^-'  y^ 

dy 

dx 

dx 

df(x,yy 

dy 

k^^'y^ 

82  INFINITESIMAL   CALCULUS,  [Ch.  IV. 

Ex.  5.    In  the  exercises  above,  test  the  first  statement  made  in  this  note. 

Note  2.     Partial  derivatives  and  the  differentiation  of  implicit  functions 
are  discussed  further  in  Chapter  VIII. 


EXAMPLES. 

N.B.  It  is  not  advisable  for  the  beginner  to  work  the  larger  part  of 
Exs.  1-8  before  proceeding  to  the  next  chapter.  Many  of  the  differentiations 
required  in  these  examples  are  far  more  difficult  than  those  that  are  commonly 
met  in  pure  and  applied  mathematics  ;  but  the  exercise  in  working  a  fair 
proportion  of  them  will  develop  a  skill  and  confidence  that  will  be  a  great 
aid  in  future  work. 

Differentiate  the  functions  in  Exs.  1-4,  6,  7,  with  respect  to  x. 

1.    (i)   (2a;-l)(;:!ic  +  4)(x2  +  ll);         (ii)   {a  +  x){h  +  x)] 
(iii)  (a  +  a;)-(&  +  x)«;     (iv)  [^  ^  "^l" ',     (v)   ^^^"  ^    ;     (vi) 


ix  +  6)«       ^  '   (1  +  xy  Va'^ 


(vii)         ^        ;         (viii)     ^^^^    ;  (ix)    Vl  +  .2+Vl-x_2 . 

V 1  +  u;2  Va+  Vx  vTTx^  -  Vl  -  x2 

(x)  ( ^  V;         (xi)  X  (a2  +  x2)  ^/aP■  -  x^. 

Vl  +  Vl  -X2/ 

2.  The  logarithms  of :    (i)  7  a:*  +  3  ic2  -  17  x  +  2  ;     (ii)  \^\  ~  ^\ ; 

^  d^  +  x^ 

(iii)  ? ;      (iv)  Jl+^!^;      (V)  J^^^±g+^. 

a  -  yJa^  -  x2  ^l-smx  \  Vl  +  x^  -  x 

3.  (i)  sin  4  x^  ;  (ii)  cos^  7  x  ;  (iii)  sec^  3  x  ;  (iv)  tan  (8  x  +  5) ; 
(v)  x^logx;  (vi)  sin^x«;  (vii)  sin  ?ix  •  sin** x  ;  (viii)  sin  (sin  x); 
(ix)  sin  (log  ?ix)  ;         (x)  log  (sin  nx). 


(iii)  log^l±|-|tan-ix. 


5.  Showthat   2){^^^^  +  ^'  +  |log(x  +  Va2  +  x^)  }  =  V^^:^ 

6.  (i)  tan-ie==;  (ii)  sin-i(cosx);  (iii)  sin(cos-ix); 

(iv)  tan-i  (w  tan  x) ;        (v)  sin-i  ^  +  ^  ^"^  ^^       (vi)  e«^sin»«rx; 
^    ^  a  +  6cosx 

(vii)  tana^;        (viii)  e^A/^-t-^- 


56.]  DIFFERENTIATION   OF  FUNCTIONS.  83 


7.  (i)   f-V^      (ii)  -e'';     (iii)  x^^     (iv)  e-^     (v)x(-");     (y\)  {x-y. 

\n )  X 

8.  Find  -y-  under  each  of  the  following  conditions  : 

(i)  ax2  +  2  hxy  +  &?/2  +  2  grx  +  2/?/  +  c  =  0 ;  (ii)  (x^  +  y-2)2  -  a^{x''  -  y^)  =0 ; 
(iii)  x^y^  +  smy  =  0]  (iv)  sin  (xj/)  =  wiaj ;  (v)  sin  x sin  y  +  sin  x  cos ?/ =  ?/ ; 
(vi)  ey  —  e'  -\-  xy  =  0]       (vii)  xM  =  y  ;        (viii)  2/e"^  =  ax"*. 

du  ^=^ 

9.  Find  ^  in  terms  of  x,  when  x  =  e  y  . 

dx 

10.  Differentiate  as  follows:  (i)  Zy'^  —  ly+W  with  respect  to  3?/; 
(ii)  4  <2  _  11  i  -f  1  with  respect  to  t  +  2;  (iii)  x  with  respect  to  sin  x  ; 
(iv)  sin  z  with  respect  to  cos  z  ;      (v)  x  with  respect  to  Vl  —  x^. 

11.  (i)  Given  ?/=:o?«-  — 7  ?<  +  2  and  w  =  2x^  +  3x  +  2,  find -^  ;  (ii)  given 
^  =  e«  +  §2  and  s  =  tan  t,  find  -p  ;  (iii)  given  v  =  >/2  (/s,  s  =  \  gt^,  find  — 
in  two  ways  ;     (iv)  2c  =  tan-i(xy),  y  =  e',  find  —  • 

12.  Compute  the  angle  at  which  the  following  curves  intersect,  and  sketch 
the  curves :  (i)  x^  -  ?/2  _  9  and  xy  =  4  ;  (ii)  x^  -\-  y^  =  25  and  Ay^  =  9x; 
(iii)  ?/2  =  8 (x  +  2)  and  y'^  +  4(x  -  I)  =  0  ;  (iv)  1/  =  3  x2  -  1  and  ?/  =  2 x2 
+  3  ;     (v)  x2  +  y2  ^  9  and  (x  -  4)2  +  ?/2  _  2  y  =  15. 

13.  A  point  P  is  moving  with  uniform  speed  along  a  circle  of  radius  a 
and  centre  O  ;  AB  is  any  diameter,  and  Q  is  the  foot  of  the  perpendicular 
from  P  on  AB.  Show  that  the  speed  of  Q  is  variable,  that  at  A  and  B  it  is 
zero,  and  at  O  it  is  equal  to  the  speed  of  P.  (The  motion  of  Q  is  called 
simple  harmonic  motion.) 

rScGGESTiON  :  Denote  angle  AOP  by  6,  and  OQ  by  x.     Then  x  =  «  cos^  ; 

hence  ^  =  - a  sin  6^-1 
dt  dt  J 

14.  Suppose,  in  Ex.  13,  the  radius  is  18  inches,  and  P  is  making  4  revolu- 
tions per  second  :  what  is  the  speed  of  Q  when  AOP  is  15°,  30"^,  45°,  60°, 
75°,  90°,  120°,  150°,  respectively  ? 


CHAPTER   V. 

SOME  GEOMETRICAL,  PHYSICAL,  AND  ANALYTICAL 
APPLICATIONS.  GEOMETRIC  DERIVATIVES  AND 
DIFFERENTIALS. 

N.B.  The  variation  of  functions,  the  sketching  of  graphs,  and  the 
determination  of  maxima  aud  minima,  which  are  discussed  in  Chapter 
VI L,  can  be  studied  before  entering  upon  this  chapter.  For  some 
reasons  it  may  be  preferable  to  do  this. 

57.  This  chapter  gives  some  practical  applications  of  the  pre- 
ceding principles  of  the  calculus.  The  applications  in  Arts.  58 
and  59  are  already  familiar  or  obvious.  Rolle's  theorem  and  the 
theorem  of  mean  value,  in  Arts.  63,  64,  are  two  of  the  most  im- 
portant theorems  in  the  calculus.  The  study  of  the  geometric 
derivatives  and  differentials,  in  Art.  67,  is  of  no  immediate  press- 
ing importance,  but  will  be  found  of  particular  interest  when 
Chapters  XIL  and  XVI.  are  taken  up.  A  glance  over  this  article, 
however,  will  serve  to  make  clearer  and  stronger  the  notions  of  a 
derivative  and  a  differential. 

58.  Slope  of  a  curve  at  any  point :  rectangular  coordinates.      It 

]ias  been  shown  in  Art.  24  that  at  a  point  (.Tj,  y^)  on  a  curve  whose 
equation  is  (1)  y  =f(x),  or  (2)  <f>{x,  y)  =  0,  the  slope  of  the  tangent 

is  -^-     [Here  -^  denotes  the  result  of  substituting  (x^,  y{)  for 
(X3/J  ax^ 

{x,  y)  in  -^  derived  from  (1)  or  (2).]     Examples  have  been  given 
ax 

in  the  preceding  articles. 

59.  Lengths  of  tangent,  subtangent,  normal,  and  subnormal,  for 
any  point  on  a  curve :  rectangular  coordinates.  Let  P  be  a  point 
(xi,  2/i)  on  the  curve  y  =f(x)  [or,  cf>(x,  y)  =  0]. 

At  P  let  the  tangent  PT  be  drawn ;  likewise  the  normal  PN 
and  the  ordinate  PM.     The  length  of  the  line  J*r,  namely,  that 

84 


67-59.] 


LENGTHS   OF  TANGENT,   ETC. 


85 


part  of  the  tangent  which  is  intercepted  between  P  and  the  a>axis, 
is  here  termed  the  length  of  the  tan- 
gent. The  projection  of  TP  on  the 
a>axis,  namely  TM,  is  called  the 
suhtangent.  The  length  of  the  line 
PN,  the  part  of  the  normal  which 
is  intercepted  between  P  and  the 
a;-axis,  is  termed  the  length  of  the 
normal.  The  projection  of  PN  on 
the  ic-axis,  namely  MN,  is  called 
the  subnormal. 

Note  1.  The  subtangent  is  measured  from  the  intersection  of  the  tangent 
with  the  X-axis  to  the  foot  of  the  ordinate  ;  the  subnormal  is  measured  from 
the  foot  of  the  ordinate  to  the  intersection  of  the  normal  with  the  x-axis. 


Let  angle  XTP  be  denoted  by  a ;  then  tan  a  = 


In  the 


triangle  TPM:   MP^y^]    TM=  y,  cot  a  =  y^^^^]    TP  =  y^c^ca 


dyi 


-MW' 


or,  TP=  -y^MP'  +  TM"  =  2/i  \  1  +  f — 


In 


the  triangle  PMN:  angle  MPN  =  a ;  MN=  y^  tan  MPN=  y^^] 

dxi 


PN  =  2/1  sec  3fPN  =y^yjl-{-  /"^Y;        fov,  PN  =  yJMP^  +  J/JV' 


=  2/r 

It  being  understood  that  y  and  -^  denote  the  ordinate  and  the 

dx 

slope  of  the  tangent  at  any  point  on  the  curve,  these  results  may 
be  written : 

snbtangent  =  y^; 
dy 

subnormal  =  y  ^5 

CTOJ 


length  of  tangent  =  2/  \'l  +  (  ^^V ; 


length  of  normal  =  2/Vl  +  (|^)* 


For,  since  tan  PTX-^^^ 


86  INFINITESIMAL   CALCULUS.  [Ch.  V. 

Note  2.     These  results  are  true,  no  matter  what  the  figure  may  be.     The 
student  is  advised  to  draw  various  figures. 

Note  3.    These  results  may  also  be  derived  by  means  of  analytic  geometry. 

dx\ 
the  equation  of  the  tangent  at  P  is  y  —  yi  =  -^  (x  —  x^)  ;  (1) 

and  the  equation  of  the  normal  at  P  is      (y  —  y{)  -^  -{-(x     Xi)  =  0.  (2) 

dxi 

Hence,  from  (1),  the  intercept  OT  =  xi  —  ?/i  — ; 

dyx 

and  from  (2)  the  intercept  ON  =^  xi  +  ?/i^- 

dxi 

The  subtangent  TM=OM-  0T  =  yi^- 

dyi 

The  subnormal  3IN=:  ON-  OMr^  yi^- 

dx\ 

Then  TP  and  FN  can  be  found  from  MP,  TM,  and  MN. 


EXAMPLES. 

N.B,  Sketch  all  the  curves  and  draw  all  the  lines  involved  in  the  follow- 
ing examples. 

1.  In  each  of  the  following  curves  write  the  equations  of  the  tangent  and 
the  normal,  and  find  the  lengths  of  the  subnormal,  subtangent,  tangent,  and 
normal,  at  any  point  (xi,  ?/i),  or  at  the  point  more  particularly  described: 
(1)  Circle  x^  +  y'^  =  25  where  x  =  -  3  ;    (2)  parabola  ?/2  =  8  x  at   x  =  2  ; 

(3)  ellipse  b^x^  +  a^y^  =  a^b'^ ;  (4)  sinusoid  y  =  smx;  (5)  exponential  curve 
y  =  e'. 

2.  Where  is  the  curve  y(x  —  2)  (x  —  3)  =-x  —  7  parallel  to  the  x-axis  ? 

3.  What  must  a^  be  in  order  that  the  curves  16  x^  +  25  y'^  =  400  and 
49  x^  +  ct^y'^  =  441  intersect  at  right  angles  ? 

X 

4.  In  the  exponential  curve  y  =  &e«  show  that  the  subtangent  is  constant 

and  that  the  subnormal  is  — • 
a 

5.  In  the  semi-cubical  parabola  Sy^  =  (x  -\-  ly  show  that  the  subnormal 
varies  as  the  square  of  the  subtangent. 

6.  In  the  hypocycloid  of  four  cusps,  x^  +  y^  =  a^ :  (1)  Write  the  equa- 
tion of  the  tangent  at  (xi,  yi)  ;  (2)  show  that  the  part  of  the  tangent 
intercepted  between  the  axes  is  of  constant  length  a ;  (3)  show  ttjat  the 
length  of  the  perpendicular  from  the  origin  on  the  tangent  at  (x,  y)  is  \/axy  ; 

(4)  if  p,  pi  be  the  lengths  of  the  perpendiculars  from  the  origin  to  the  tangent 
and  normal  at  any  point  on  the  curve,  4p"2  -j-  pi^  =  a^. 


59.]  EXAMPLES.  87 

7.  In  the  parabola  x^  -\-  y^  —  a^,  write  the  equation  of  the  tangent  at 
any  point  (xi,  j/i),  and  show  that  the  sum  of  the  intercepts  made  on  the  axes 
by  this  tangent  is  constant.  Show  that  this  curve  touches  the  axes  at  (a,  0) 
and  (0,  a). 

8.  In  the  cycloid  x  =  a{d  —  sin  ^),  y  =  a{l  —  cos  6):  (1)  Calculate  the 
lengths  of  the  subnormal,  subtangent,  normal,  and  tangent  at  any  point 
(x,  y)  ;  (2)  show  that  the  tangent  at  any  point  crosses  the  y-axis  at  the  angle 

-;  (8)  show  that  the  part  of  the  tangent  intercepted  between  the  axes  is 

ad  cosec  — 

2 

9.  In  the  hyperbola  xy  =  d^ :    (1)  Show  that  for  any  point  (a;,  y)  on 

the  curve  the  subnormal  is  —  ^   and  the  subtangent  is  —  a; ;  (2)  find  the 

X-  and  ?/-intercepts  of  the  tangent  at  any  point  (aci,  j/i),  and  thence  deduce  a 
method  of  drawing  the  tangent  and  normal  to  the  curve  at  any  point  on  it. 
Show  that  the  product  of  these  intercepts  is  4  c'^. 

10.  In  the  semi-cubical  parabola  ay'^  =  x^,  show  that  the  length  of  the 
subtangent  for  any  point  (x,  y)  is  f  x ;  thence  deduce  a  way  of  drawing  the 
tangent  and  the  normal  to  the  curve  at  any  point  on  it. 

Q 

11.  Show  that   the   parabola   ic^  =  4  ?/   intersects    the   witch    y  = 

at  an  angle  tan"!  3  ;  i.e.  ir  33'  54".  ^^  +  ^ 

12.  Find  at  what  angles  the  parabola  y'^  —  2ax  cuts  the  folium  of  Descartes 
x^  -^  if  =  3  axy. 

13.  In  the  curve  x"*?/"  =  «"*+"  show:  (1)  That  the  subtangent  for  any 
point  varies  as  the  abscissa  of  the  point ;  (2)  that  the  portion  of  the  tangent 
intercepted  between  the  axes  is  divided  at  its  point  of  contact  into  segments 
which  are  to  each  other  in  the  constant  ratio  m  -.n',  (3)  thence,  deduce  a 
method  of  drawing  the  tangent  and  the  normal  at  any  point  on  the  curve. 
(The  curves  x'"v/"  =  «"»+",  obtained  by  giving  various  values  to  m  and  n,  are 
called  adiahatic  curves.  Instances  of  these  curves  are  given  in  Exs.  9,  10, 
and  in  the  parabolas  in  Exs.  11,  12.) 

14.  Show  that  all  the  curves  obtained  by  giving  different  values  to  n  in 
(-)  +(-)  =2?  touch  one  another  at  the  point  (a,  6).  Draw  the  curves  in 
which  (a,  6)  is  (4,  7),  n  =  1,  n  =  2. 

15.  Show  that  the  tangents  at  the  points  where  the  parabola  ay  =  x^ 
meets  the  folium  of  Descartes  ji^  +  y^  =  S  axy  are  parallel  to  the  oj-axis,  and 
that  the  tangents  at  the  points  where  the  parabola  y^  =  ax  meets  the  folium 
are  parallel  to  the  ?/-axis.  Make  figures  for  the  curves  in  which  a  =  1  and 
a  =  4. 


88  INFINITESIMAL   CALCULUS.  {Cn.Y. 

60.   Slope  of  a  curve  at  any  point :  polar  coordinates.     Let  CM 

be  a  curve  whose  equation  is 
r=f(e),  [or  <^(r,  ^)=0],  and 
P  be  any  point  on  it  having 
coordinates  y\,  6i,  with  reference 
to  the  pole  0  and  the  initial 
line  OL.  Draw  OP;  then 
OP=r^,  and  angle  LOP  =6^. 
Through  P  and  Q  (a  neigh- 
bouring   point    on    the    curve), 

draw  the  chord  TPQ,  and  draw  OQ.     From  P  draw  PR  at  right 
angles  to  OQ. 

Let  angle         POQ  =  A(9i,  and  OQ  =  ri  +  A?'i ; 

then        PE  =  ?\  sin  A^i,  and  i?Q  =  ?'i  -|-  A?'i  —  r^  cos  A^j. 

The  angle  between  the  radius  vector  drawn  to  any  point  P  and 
the  tangent  at  P  is  usually  denoted  by  i/^.     Since 

x}/  =  lim^g^..o  angle  BQP, 

then,  using  the  general  coordinates  r,  6,  instead  of  r^,  ^j. 


MP 

tani/.  =  lim^g^  — 


=  lini 


sin  A^ 


AS^ 


r  +  Ar  —  r  cos  A^ 


On  replacing  cos  A^  by  its  equal,  1  —  2  sin^  ^  A^,  and  dividing 
numerator  and  denominator  by  A^,  this  becomes 

sin  A^ 
r 

tan  i/f  =  lim^^^ 


Ar  ,        .     .  ^  ,    sm  "I  A^      dr 


That  is,  tantl/^?-^.  (1) 

The  angle  between  the  initial  line  and  the  tangent  at  P  is 
usually  denoted  by  <^. 


LENGTHS   OF  TANGENT,   ETC, 


),61.] 

It  is  apparent  from  Fig.  17  that 

«|)  =  t|f  +  0. 


89 


(2) 


Note.  Results  (1)  and  (2)  are  true  for  all  polar  curves,  whatever  the 
figure  may  be.    The  student  is  advised  to  draw  various  figures. 

61.  Lengths  of  the  tangent,  normal,  subtangent,  and  subnormal,  for 
any  point  on  a  curve :  polar  coordinates. 

In  Fig.  18  0  is  the  pole  and  OL  is  the  initial  line.  At  P  any 
point  (?*!,  Oi),  on  the  curve  CR,  whose 
equation  is  r=f(0),  [or  cf>(r,  ^)  =  0], 
let  the  tangent  PT  and  the  normal 
PN  be  drawn.  Produce  them  to 
intersect  NT,  which  is  drawn  through 
0  at  right  angles  to  the  radius  vector 
OP. 

The  length  of  the  line  PTis  termed 
the  length  of  the  tangent  at  P;  the 
projection  of  PT  on  NT,  namely  OT, 
is  called  the  polar  subtangent  for  P; 
the  length  of  PN  is  termed  the 
length  of  the  normal  at  P;  the  projec- 
tion of  PN  on  NT,  namely  ON,  is  called  the  polar  subnormal  for  P. 

Note.  In  Art.  59  the  line  used  with  the  tangent  and  the  normal  is  the 
X-axis.  Here  the  line  so  used  is  not  the  initial  line,  but  the  line  drawn 
through  the  pole  at  right  angles  to  the  radius  vector  of  the  point. 

In  the  triangle    OPT: 

0T=  OP  tan  OPT: 


Fig.  18. 


90  INFINITESIMAL    CALCULUS.  [Ch.  V. 

i.e.  (on  removing  the  subscripts  from  the  letters) 
polar  subtan^ent  =  r  tan  if/  =  r^—; 
also,  TP=OPseGOPT', 


i.e.  polar  tangent  length  =  r  sec  i/^  =  r  ^'1  +  r2  ( — 

In  the  triangle  OPN : 

angle  iV^PO  =  90-1/^; 

ON=OPta,nNPO', 
i.e.  polar  subnormal  =  r  cot  j/^  =  — ^ ; 

also,  iV7^=  OP  sec  N^PO; 

i.e.  polar  normal  length  =  r  cosec  i/^  =-y/i'2  +  l-~\  . 


["or  :  ^P  =  VOP'  +  OiV^'  =  yj  r^  +  (— Y-1 


Note.     In  Fig.  18  r  increases  as  6  increases;  accordingly  —  is  positive, 

dd  ^*' 

and  hence  the  subtangent  is  positive.     Thus  when  —  is  positive,  the  sub- 

dr 

tangent  is  measured  to  the  right  from  an  observer  at  0  looking  toward  P. 

When  r  decreases  as  6  increases,  and  thus  —  is  negative,  the  subtangent  is 

dr 
measured  to  the  left  of  the  observer  looking  toward  P  from  O.    The  student 
is  advised  to  construct  figures  for  the  various  cases. 


EXAMPLES. 

N.B.    In  the  following  examples  make  figures,  putting  a  =  4,  say.    Apply 
the  general  results  found  in  these  examples  to  particular  concrete  cases,  e.g. 

a  =  6  and  e  =  -,  a  =  2  and  d  =  — ,  etc.    The  angle  6,  as  used  in  the  equa- 
tions of  the  curves,  is  expressed  in  radians. 


62.]  APPLICATIONS  INVOLVING   RATES.  91 

1.  In  the  following  curves  calculate  the  lengths  of  the  subnormal,  sub- 
tangent,  normal,  and  tangent,  at  any  point  (r,  ^)  :  (1)  7"he  spiral  of 
Archimedes  r  =  ad ;  (2)  the  parabolic  spiral  or  lituiis  r^  =  a^d  {i.e. 
r  =  ad^)  ;  (3)  the  hyperbolic  spiral  (or  the  reciprocal  spiral)  rd  =  a; 
(4)  the  general  spiral  r  =  ad'K  (The  preceding  spirals  are  special  cases 
of  this  spiral.) 

2.  From  the  results  in  Ex.  1  deduce  simple  geometrical  methods  of 
drawing  tangents  and  normals  to  the  spirals  in  (1),  (2),  (3), 

3.  Do  as  in  Exs.  1,  2,  for  the  logarithmic  spiral  r  =  e«^.  In  this 
curve  each  of  the  lengths  specified  varies  as  the  radius  vector. 

4.  (a)  In  the  spiral  of  Archimedes  r  =  ad,  show  that  tan  \}/  =  6.  Find 
t//  and  0  in  degrees  when  angle  TOP  (Fig.  17)  =  40°,  and  when  TOP  =  70°. 
(6)  In  the  curve  r  =  4:6,  find  yp  and  0  when  r  =  2. 

5.  («)  In  the  logarithmic  spiral  r  =  ce«^,  show  that  \}/  is  constant. 
This  spiral  accordingly  crosses  the  radii  vectores  at  a  constant  angle,  and 
hence  is  also  called  the  equiangular  spiral,  (b)  Show  that  the  circle  is  a 
special  case  of  the  logarithmic  spiral,  and  give  the  values  of  ^  and  a  for 
this  case. 

6.  In  the  parabola  r  =  asec2-,  show  that  </>  + 1/' =  tt.  Make  a  prac- 
tical  application  of  this  fact  to  drawing  tangents  and  normals  of  this  curve. 

7.  In    the    cardioid   r  =  a(l  —  cos  6),    show    that  0  =— ,  V'  = -,  sub- 

6  6  2  2 

tangent  =  2  «  tan  -  sin^  -.    Apply  one  of  these  facts  to  drawing  the  tangent 

and  normal  at  a  point  on  the  curve. 

62.  Applications  involving  rates.  Applications  of  this  kind 
have  already  been  made  in  Arts.  26,  37.  Rates  and  differentials 
have  been  discussed  in  Arts.  25-27.  It  has  been  seen,  Art.  26, 
Eq.  (1),  that  if  y  =f(x),  then 

'      dt      -^  ^^  Ut      dx    dt 

In  words,  the  rate  of  change  of  a  function  of  a  variable  is  equal 
to  the  product  of  the  derivative  of  the  function  with  respect  to 
the  variable  and  the  rate  of  change  of  the  variable.  The  following 
principles,  which  are  proved  in  mechanics,  will  be  useful  in  some 
of  the  examples :  (a)  If  a  point  is  moving  at  a  particular  moment 

in  such  a  way  that  its  abscissa  x  is  changing  at  the  rate  — ,  and 

dt 


92  INFINITESIMAL   CALCULUS.  [Ch,  V. 

its  ordinate  y  is  changing  at  the  rate  ~^',  and.  if  —  denote  its  rate 

(It  dt 

of  motion  along  its  path  at  that  moment,  then 
'dsY     fdx\^ 


dtj      \dt  ' 


(ij 


(6)  If  a  point  is  moving  in  a  certain  direction  with  a  velocity 
V,  the  component  of  this  velocity  in  a  direction  inclined  at  an 
angle  a  to  the  first  direction,  is  v  cos  a. 

For  instance,  if  a  point  is  moving  so  that  its  abscissa  is  increasing  at  tlie 
rate  2  feet  per  second  and  its  ordinate  is  decreasing  at  tlie  rate  8  feet  per 
second,  it  is  moving  at  the  rate  V2'^  +  3'^,  i.e.  Vl8  feet  per  second.  Again, 
if  a  point  is  moving  at  the  rate  of  6  feet  per  second  in  a  direction  inclined 
60°  to  the  X-axis,  the  component  of  its  speed  in  a  direction  parallel  to  the 
aj-axis  is  6  cos  60°,  i.  e.  3  feet  per  second,  and  the  component  parallel  to  the 
y-axis  is  6  cos  30°,  i.e.  5.196  feet  per  second. 


EXAMPLES. 
N.B.    Make  figures. 

1.  If  a  particle  is  moving  along  a  parabola  y^  =Sx  at  a  miiform  speed  of 
4  feet  per  second,  at  what  rates  are  its  abscissa  and  its  ordinate  respectively 
increasing  as  it  is  passing  through  the  point  (cc,  y)  and  x  has  successively  the 
values  0,  2,  8,  16  ? 

2.  A  particle  is  moving  along  a  parabola  y^  =  ix,  and,  w^hen  x  =  i,  its 
ordinate  is  increasing  at  the  rate  of  10  feet  per  second  :  find  at  what  rate  its 
abscissa  is  then  changing,  and  calculate  the  speed  along  the  curve  at  that 
time. 

3.  A  particle  is  moving  along  the  hyperbola  xy  =  25  with  a  uniform  speed 
10  feet  per  second  :  calculate  the  rates  at  which  its  distances  from  the  axes 
are  changing  when  it  is  distant  1  unit  and  10  units  respectively  from  the 
y-axis. 

4.  A  vertical  wheel  of  radius  3  feet  is  making  25  revolutions  per  second 
about  an  axis  through  its  centre  :  calculate  the  vertical  and  the  horizontal 
components  of  the  velocity,  (1)  of  a  point  20°  above  the  level  of  the  axis; 
(2)  of  a  point  65°  above  the  level  of  the  axis. 

5.  A  point  is  moving  along  a  cubical  parabola  y  =  x^ :  find  (1)  at  what 
points  the  ordinate  is  increasing  12  times  as  fast  as  the  abscissa  ;  (2)  at  what 
points  the  abscissa  is  increasing  12  times  as  fast  as  the  ordinate  ;  (3)  how 
many  times  as  fast  as  the  abscissa  is  the  ordinate  growing  when  oj  =  10  ? 


63.]  BOLLE'S   THEOREM.  93 

63.   RoUe's  Theorem. 

Note  1,  Progressive  and  regressive  derivative.  In  Art.  22  the  derivative 
of  /(x)  v\ras  defined  as 

,^^^jj^±^^,fM,       (1) 

The  process  of  evaluating  (1)  is  equivalent 
to  the  geometrical  process  of  revolving  the 
chord  PQ  of  the  curve  y  =f{x)  about  P  until 
Q  coincides  with  P,  and  thus  PQ  becomes  the 
tangent  PT.  If  in  this  curve  a  chord  PB  be 
drawn,  and  BP  be  revolved  about  P  until  B 
coincides  with  P,  then  BP  will  finally  take  the 
position  PT.     The  slope  of   the  tangent  ob-  Fig.  19. 

tained  by  thus  revolving  BP  is  evidently 

/(x)-/(x-Ax).                       /rx-Ax)-/(x) 
"iiiAa;^  ^  ,   i.e.  iim^j.^ ITAx ^  '^ 

It  is  customary  to  call  (1)  the  progressive  derivative,  and  (2)  the  regressive 
derivative.  In  general  these  derivatives  are  equal ;  that  is,  in  general  the 
tangent  on  the  representative  curve  is  the  same,  whether  the  secant  which  is 
revolved  until  it  assumes  a  tangential  position  be  drawn  forward  or  backward 
from  the  point  under  consideration.  In  some  cases,  however,  these  derivatives 
are  not  equal ;  such  a  case  is  represented  at  P  on  Fig.  21  c,  where  the  two 
revolving  secants  give  two  different  tangents.  In  such  a  case  the  derivative 
is  discontinuous  at  P,  for  its  value  suddenly  changes  from  the  slope  of  TP 
to  the  slope  of  LP. 

Theorem.  If  a  function  f(x)  and  its  derivative  f'(x)  are  continu- 
ous for  all  values  of  x  between  a  and  b,  and  if  f{a)  =f(b),  then 
f'(x)  =  0  for  at  lea^t  one  value  of  x  between  a  and  b. 

Following  is  a  geometrical  proof*  and  representation  of  this 
theorem.  Let  the  curve  MN  (Figs.  20  a,  b,  c)  represent  the 
function  f(x). 

At  M  and  ^  let  x=a  and  x  =  b  respectively.  Since  the  ordi- 
nates  AM  and  BN  are  equal,  it  is  evident  that  there  must  be  at 
least  one  point  between  M  and  N  where  the  function  ceases  to 
increase  and  begins  to  decrease,  or  ceases  to  decrease  and  begins 


*  An  analytical  discussion  will  be  found  in  the  collateral  reading  suggested 
in  Note  2,  Art.  64. 


94 


IN  FINITE  SIM  A  L   CALC  UL  US. 


[Ch.  V. 


to  increase.  There  may  be  several  such  points,  as  in  Fig.  20  c. 
But  at  such  a  point,  for  instance  F,  or  F^,  or  F^,  or  1\,  where  x=Xi 
say, /'(a^i)  =  0.     (If  f(x)  is  constant,  then  f'(x)  =  0  at  every  point.) 


Fig.  20  a. 


Fig.  20  6. 


Fig.  20  c. 


A  special  case  of  this  theorem  is  that  in  which  /(a)  =  0  and 
f(b)  =  0.  The  student  may  construct  the  figure  for  himself  by 
merely  moving  OX  to  the  position  MN.  For  an  application  to 
the  theory  of  equations  and  for  the  corresponding  algebraic 
statement  of  the  theorem,  see  Art.  66  B. 

Note  2.  The  necessity  of  the  condition  relating  to  continuity  is  evident 
from  Figs.  21  a,  6,  c,  d. 


Fig.  21  a. 


Fig.  21  6. 


Fig.  21  c. 


Fig.  21  d. 


For  a  value  of  x  between  x  =  a  and  x  =  h  :  in  Fig.  21  a,  f(x)  is  infinite  ; 
in  Fig.  21  6,  /(x)  is  discontinuous  ;  in  Fig.  21  c,  f{x)  is  discontinuous  ;  in 
Fig.  21  d,  f{x)  is  infinite. 

64.  Theorem  of  mean  value.  If  a  function  f(x)  and.  its  derivative 
f'(x)  are  continuous  for  all  values  of  x  from  x  =  a  to  x  =  b,  then 
there  is  at  least  one  value  of  x,  say  x^,  between  a  and  b  such  that 

J{b)-f(a)_ 


64.] 


THEOREM  OF  MEAN   VALUE. 


95 


Following  is  a  geometrical  proof*  and  explanation  of  this 
theorem. 

Let  the  curve  MN  (Fig.  22  a  or  Fig.  22  b)  represent  the  func- 
tion f(x).     Draw  the  ordinates^lP  and  BQ  at  A  and  B,  where 


Fig.  22  a. 


Fig.  22  6. 


x  =  a  2iud  x=b  respectively.     Draw  PQ  and  draw  PR  parallel  to 
OX.     Then  AP=f(a),     BQ=f(b). 


Hence 
and 


BQ=f(b)-f(a), 
^     PR  b-a 


Now  the  chord  PQ  and  the  tangent  ST  drawn  at  some  point  V 
(or  Vi  and  F2)  between  P  and  Q  evidently  must  be  parallel.  At 
Flet  x=Xi,  Xi  thus  being  between  a  and  b  ;  then  tan  RPQ=f'{x^. 


Hence 


b-a 


(1) 


Since  x^  is   between  a  and    b,  Xi  =  a  +  6(b  —  a),  in  which  0 

denotes  some  number  between  0  and  1  (i.e.  0  <  ^  <  1).  Accord- 
ingly, theorem  (1)  may  be  expressed 

/(6)=/(a)  +  (6-a)/Ta  +  e(6-a)].  (2) 

If  b  —  a  =  h,  then  b  =  a  -\- h,  and  (2)  is  written 

f(a  -^h)  =  f{a)  +  hf  (a  -f-  efe) .  (3) 


*  For  an  analytical  deduction  of  the  theorem  of  mean  value  from  Rolle' 
theorem,  see  Art.  176. 


96  INFINITESIMAL   CALCULUS.  [Ch.  V. 

Eesult  (3)  has  important  applications.  It  is  very  useful  for 
finding  an  approximate  value  of  /(a  +  h)  when  f{x),  a,  and  li,  are 
given.  A  closer  approximation  to  the  value  of  f{a  -f  h)  can  be 
found  by  Taylor's  formula,  Art.  176. 

Note  1.  The  necessity  for  the  condition  relating  to  continuity  can  be 
made  evident  by  figures  similar  to  Figs.  21  a,  i,  c,  d. 

Note  2.  References  for  collateral  reading  on  Rollers  theorem  and  the 
theorem  of  mean  value:  McMahon  and  Snyder,  Diff.  Cal.,  Arts.  59,  66; 
Lamb,  Calculus,  Arts.  48,  49,  56  ;  Gibson,  Calculus,  §§  72,  73 ;  Harnack, 
Calculus,  Art.  22  ;  Echols,  Calculus,  Chap.  V.  The  last  mentioned  text  has 
a  particularly  full  and  valuable  account  of  these  theorems. 

EXAMPLES. 

1.  Find  by  relation  (3)  an  approximate  value  of  sin  32°  20'  taking  a=32°  : 
(1)  putting  6  =  0,  (2)  putting  6  =  1;  and  compare  the  calculated  results 
with  that  given  in  the  tables. 

2.  If  /(x)  =  2x^  —  x  +  5,  find  what  6  must  be  in  order  that  relation 
(3)  be  satisfied :   (1)  when  a  =  3  and  h  =  1 ;  (2)  when  a  =  10  and  h  =  2. 

3.  Show  that  for  any  quadratic  function,  say  /(a?)  =  Ix^  +  mx  +  n, 
/(a  4-  h)  will  be  obtained  by  putting  ^  =  i  in  relation  (3) .  What  geometrical 
property  of  the  parabola  corresponds  to  this  ?     (Deduce  the  value  of  6.) 

4.  If  f(x)  =  x^,  find  what  6  must  be  in  order  that  relation  (3)  be  satisfied 
when  a  =  S  and  h  =  1.  What  problem  in  connection  with  the  cubical 
parabola  y  =  x^  is  the  correlative  of  this  ? 

65.   Small  errors  and  corrections :  relative  error. 

If  2/=/(^),  (1) 

then  by  Art.  27,  dy  =  f'(x)  •  dx,  (2) 

in  which  dx  is  an  assigned  change  in  x.  It  has  been  seen  (Note 
3,  Art.  27)  that  d.y  is  approximately  the  change  in  y  due  to  dx. 
An  important  practical  application  may  be  made  of  this  principle. 
For  it  follows  that  if  dx  be  regarded  as  a  small  error  in  the 
assigned  or  measured  value  of  x,  then  dy  is  an  approximate  value 
of  the  consequent  error  in  y. 

The  ratio  ^  ov -^  •  dx  (3) 

is,  approximately,  the  relative  error  or  the  proportional  error^  i.e. 
the  ratio  of  the  error  in  the  value  to  the  value  itself. 


65.]  SMALL   ERBORS.  97 

The  approximate  values  of  the  correction  and  relative  error  may  also  be 
deduced  from  the  theorem  of  mean  value.  For,  if  y  =f{x),  and  Ax  be  an 
error  in  x,  then /(x  +  Ax)  - /(x)  is  the  error  in  y,  i.e.  the  correction  that 
must  be  applied  to  y.    Now  by  (3)  Art.  64,  on  putting  a  =  x  and  h  =  Ax, 

/(x  +  Ax)  - /(x)  = /' (x  +  0  .  Ax)  .  Ax. 

Hence,  on  denoting  the  error  in  y  by  Ay, 

Ay  =/'(x)  •  Ax  approximately. 

Ay      f'(x^ 
From  this  the  relative  error  is,  approximately,  — ^  =''—^-^-  Ax.  (4) 

EXAMPLES. 

1.  The  side  a  of  a  square  is  measured,  but  there  is  a  possible  error 
Aa  :  find  approximately  the  error  in  the  calculated  value  of  the  area.  Let 
A  denote  the  area.    Then  A  =  a- ;  whence  A^  =  2  a  •  Aa  approximately. 

2.  If  the  measured  length  of  the  side  is  100  inches  and  this  be  correct 
to  within  a  tenth  of  an  inch,  find  an  approximate  value  of  the  possible  error 
in  the  computed  area,  and  an  approximate  value  of  the  relative  error. 

In  this  case,  approximately,  Aa  =  2  x  100  x  .1  =  20  square  inches.     The 

relative  error  is,  approximately,  -^  or  —  ;  that  is,  20  square  inches  in 
10,000  square  inches,  or  1  square  inch  in  500  square  inches. 

3.  A  cylinder  has  a  height  h  and  a  radius  r  inches ;  there  is  a  possible 
error  Ar  inches  in  r  :  find  by  the  calculus  an  approximate  value  of  the  possible 
error  in  the  computed  volume.  If  A  =  10  inches  and  the  radius  is  8  ±  .05 
inches  calculate  approximately  the  possible  error  in  the  computed  volume 
and  the  relative  error  made  on  taking  r  =  8  inches. 

4.  Find  approximately  the  error  made  in  the  volume  of  a  sphere  by 
making  an  error  Ar  in  the  radius  r.  The  radius  of  a  sphere  is  said  to  be  20 
inches :  give  approximate  values  of  the  errors  made  in  the  computed  surface 
and  volume,  if  there  be  an  error  of  .1  inch  in  the  length  assigned  to  the  radius. 
Also  calculate  the  relative  errors  in  the  radius,  the  surface,  and  the  volume, 
and  compare  these  relative  errors. 

5.  Two  sides  of  a  triangle  are  20  inches  and  35  inches.  Their  included 
angle  is  measured  and  found  to  be  48°  30'.  It  is  discovered  later  that  there 
is  an  error  of  20'  in  this  measurement.  Find,  by  the  calculus,  approximately 
the  error  in  the  computed  value  of  the  area  of  the  triangle.  Compare  the 
relative  errors  in  the  angle  and  in  the  area. 

6.  The  exact  values  of  the  errors  in  the  computed  values  in  Exs.  1-4 
happen  to  be  easily  found.  Calculate  these  exact  values,  and  compare  with 
the  approximate  values  already  obtained. 


98  INFINITESIMAL    CALCULUS.  [Ch.  V. 

7.  (1)  Two  sides,  a,  &,  of  a  triangle  are  measured,  and  also  the  included 
angle  C :  show  that  the  approximate  amount  of  the  error  in  the  computed 
length  of  the  third  side  c  due  to  a  small  error  A  C  made  in  measuring  C,  is 

ah  sin  C 


Va2  +  62  -  2  a& 


AC. 


(2)  Calculate  the  approximate  error  in  the  computed  value  of  the  third 
side  in  Ex.  5. 

66.   Applications  to  algebra. 

A.  If  f(x)  is  a  rational  integral  function*  of  x,  and  {x  —  ay  is 
a  factor  of  f(x),  then  (x  —  ay~^  is  a  factor  of  f  (x). 

Let  f{x)  =  (x  —  ay<f)(x). 

Then  /'  (x)  =  r  (x  -  ay-'  cf>  (x)  -\- (x  -  ay  <f>'  (x) 

=  {x  —  a)'-i  [rcl>{x)  +  (x--  a)cf>'{x)^. 

It  follows  from  this  theorem  that  if  f{x)  is  a  rational  integral 
function  of  x,  and  a  is  an  r-tuple  (or  rfold)  root  of  the  equation 
f(x)  =0,  then  a  is  an  (r—  Vytuple  root  of  the  equation  f'(x)  =  0. 
This  theorem  may  be  employed  in  finding  the  multiple  roots  of 
an  equation. 

Ex.  1.    Solve  aj3  —  2  aj2  —  15  X  4-  36  =  0  by  trying  for  equal  roots. 

The  derived  equation  is      3  a;^  —  4  aj  —  15  =  0, 
i.e.  (3  a;  +  5)  (x  -  3)  =  0. 

Trial  will  show  that  (x  -  3)2  is  a  factor  of  x^  -  2  x2  -  15  x  +  36,  and  the 
first  equation  is  (x  —  3)2  (x  +  4)  =  0.     The  roots  are  thus  :  3,  3,   —  4. 

Note.  The  multiple  roots  of  /(x)  =  0  loill  be  revealed  on  finding  the 
highest  common  factor  of  f{x)  and  /'(x). 

Ex.  2.    Solve  the  following  "equations : 

(1)  3  x3  +  4  x2  -  X  -  2  =  0 

(2)  4  x3  +  16  x2  +  21  X  +  9  =  0 

(3)  X*  -  11  x3  +  44  x2  -  76  X  +  48  =  0 

(4)  8  x*  4-  4  x3  -  62  x2  -  61  X  -  15  =  0 

(5)  x6  +  x4  -  13  x»  -  x2  +  48  X  -  36  =  0. 

Ex.  3.   Find  the  condition  that  x"  —  px^  +  r  =  0  may  have  equal  roots. 

*  A  rational  integral  function  of  x  is  a  function  in  which  x  has  only  posi- 
tive integral  exponents  and  does  not  appear  in  the  denominator  of  a  fraction  ; 
e.g.  x2  —  ^  X  +  2,  ax»  +  &x"-i  +  •••  +  mx  +  p,  if  n  is  a  positive  integer. 


66,  67.] 


GEOMETRIC  DERIVATIVES, 


99 


B.  An  important  application  of  Boilers  Theorem  may  be  made  to 
the  theory  of  equations.  According  to  the  theorem,  geometrically, 
the  slope  of  a  curve  y=f(x)  is  zero  once  at  least,  between  the 


Fig.  23  a. 


Fig.  23  &. 


points  where  the  curve  crosses  the  a^axis.  Hence,  at  least  one 
real  root  of  the  equation  f'(oc)  =0  lies  between  any  two  real  roots 
of  the  equation  fioc)  =  0.  (In  the  theory  of  equations  this  is 
called  Rolle's  Theorem.*) 

Note.  According  to  this  principle  r  real  roots  of  an  equation  f(x)  =  0 
have  at  least  (r  —  1)  roots  oi  f'{x)  =  0  between  them.  Now,  if  the  r  roots 
coalesce  and  thus  make  an  r-tuple  root,  the  (r  —  1)  roots  must  also  coalesce 
and  thus  make  an  (r  —  l)-tuple  root  oif'(x)  =  0.     (Compare  A  above.) 

Ex.  Verify  Rolle's  Theorem  in  each  of  the  following  equations  /(x)  =  0  ; 
also  sketch  the  curve  y  =  f(x)  : 


(1)  x^  +  x-Q  =  0 


(2)  x3  +  2  x2  -  5  a;  -  6  =  0. 


67.   Geometric  derivatives  and  differentials. 

(a)  Derivative  and  diflferential  of  an 
area :  rectangular  coordinates.    Let  PQ 

be  an  arc  of  the  curve  y  =  f{x) .  Take 
any  point  on  P§,  F(x,  y)  say,  and  take 
T(x  +  Ax,  y  +  Ay) .  Construct  the  rec- 
tangles VN  and  TM  as  shown  in  Fig.  24. 
Draw  the  ordinate  BP^  and  let  the  area  of 
BPVM  be  denoted  by  A\  then  the  area 
of  JfFTiV  may  be  denoted  by  A^. 


Now, 


Fig.  24. 
rectangle  VN<MVTN<  rectangle  MT  \ 


y  •  Ax  <      A^ 


+  Ay)  Ax. 


Hence,  on  division  by  Ax,  y  <      -^     <  y  +  Ay. 

Ax 


a) 


*  After  Michel  Bolle  (1652-1719). 


100  INFINITESIMAL  CALCULUS.  [Ch.  V. 

On  letting  Aa:  approach  zero,  these  quantities  (Arts.  18,  22,  23)  approach 

A 

the  values  y,  — ,  i/,  respectively. 
dx 

That  is,  the  derivative  of  the  area  BPVM  with  respect  to  the  abscissa 
X  of  F,  is  the  measure  of  the  ordinate  of  V.  On  denoting  this  measure  by  ?/, 
result  (2)  means  (Art.  26)  that  the  area  BPVM  is  increasing  y  times  as  fast 
as  the  abscissa  of  V.    From  (2)  it  follows  by  Art.  27  that 

dA  =  y  •  doc,  (3) 

That  is,  the  differential  of  the  area  BPVM  is  the  area  of  a  rectangle 
whose  height  is  the  ordinate  MV  and  whose  base  is  dx^  the  differential  of  the 
abscissa  of  V. 

Ex.  1.  Find  the  derivative  of  the  area  between  the  ic-axis  and  the  curve 
y  =  x^  with  respect  to  the  abscissa :  (a)  at  the  point  whose  abscissa  is  2  ; 
(&)  at  the  point  whose  abscissa  is  4. 

(a)    —  =  ?/,  (where  x  =  2,)  =  2^  =  8.  (4) 

dx 

(6)    —  =  y,  (where  a;  =  4,)  ==  4^  =  64.  (5) 

These  results  mean  that,  if  an  ordinate,  like  VM  in  the  figure,  is  moving 
to  the  right  or  left  at  a  certain  rate,  the  area  of  the  figure  bounded  on  one 
side  by  that  ordinate  is  changing,  in  case  (a)  at  8  times  that  rate,  and  in 
case  (6)  at  64  times  that  rate. 

Ex.  2.  Find  the  differentials  in  Ex.  1  (a)  and  (6),  when  dx  =  A  inch. 
Show  these  differentials  on  a  drawing. 

By  (8),  (4),  and  (5),  in  case  (a),  dA  =  .8  square  inch;  in  case  (&) 
dA  =  6.4  square  inches. 

Note.  The  area  .8  square  inch  is  nearly  the  actual  increase  in  ai-ea 
between  the  curve  and  the  ic-axis  when  the  ordinate  moves  from  a;  =  -2  to 
oj  =  2.1 ;  and  6.4  square  inches  is  nearly  the  increase  in  this  area  when  the 
ordinate  moves  from  x  =  4^  to  x  =  4A.  These  increases  are  calculated  in 
Ex.  16,  Art.  111. 

It  is  evident  that  the  smaller  dx  is  taken,  the  more  nearly  will  the  differen- 
tial of  the  area  become  equal  to  the  actual  increase  of  the  area  between  the 
curve  and  the  x-axis. 

Ex.  3.  Show  that  the  y-derivative  of  an  area  between  the  curve  and  the 
y-axis  is  x.  Thence  deduce  that  the  ^-differential  of  this  area  is  x  dy,  and  make 
a  figure  showing  this  differential  area. 


67.] 


GEOMETRIC  DERIVATIVES. 


101 


Ex.  4.   In  the  case  of  the  cubical  parabola  y  —  x^  find  —  and  — ;  then 

dx  dy 

calculate  the  differential  of  the  area  between  this  curve  and  the  x-axis  at  the 
point  (2,  8) ,  taking  dx  =  .2.  Also  calculate  the  differential  of  the  area  between 
this  curve  and  the  y-axis  at  the  same  point,  taking  dy  =  .2.  Show  these 
differentials  in  a  figure. 


(&)  Deriyatiye  and  differential  of  an  area :  polar  coordinates.    Let 

PQ  be  an  arc  of  the  curve  /(r,  6)  =  0.  On 
PQ  take  any  point  F(r,  6),  and  take  the 
point  TF(r  +  Ar,  0  +  A^).  About  O  describe 
a  circular  arc  VN  intersecting  OW  in  N,  and 
describe  a  circular  arc  WM  intersecting  OV 
in  31.  Then  NW  =  Ar,  and  VOW  =  Ad. 
Also  (PI.  Trig.,  p.  175),  area  sector  VON  = 
I  r^Ad,  and  area  sector  310  W=:  i  (r+AryAd. 

Draw  OP.  Let  the  area  of  POV  be 
denoted  by  A ;  then  the  area  of  VO  W  may 
be  denoted  by  AA. 

Now,  area  VOy^  <  area  VO  W  <  area  310  W\ 

i.e.  ^  r^Ad  <AA<\{r  +  ArYAd. 

AA 


f^< 


A9 


K»*  +  Ar)2. 


On  letting  Ad  approach  zero,  these  quantities  (Arts.  18,  22,  23)  approach 
the  values  ^    ,    ^  j^ 


I  r2,  — ,   ^  r2,  respectively. 
dd 


(1) 


Result  (1)  means  that,  if  the  radius  vector  is  revolving  at  a  certain  rate, 
the  area  passed  over  by  the  radius  vector,  when  its  length  is  r,  is  increasing 
at  a  rate  which  is  \  r^  {i.e.  the  number)  times  the  rate  of  revolution. 

It  follows  from  (1)  and  Art.  27  that 


dA  =  ir^d9, 


(2) 


Ex.  5.  Show  that  in  the  case  of  the  circle  the  differential  of  the  area  swept 
over  by  a  revolving  radius  is  the  additional  area  passed  over. 

Ex.  6.  In  the  spiral  of  Archimedes  r  =  2  0  find  the  derivative  of  the  area 
swept  over  by  the  radius  vector,  with  respect  to  0.  Calculate  the  differential 
of  this  area  when  :  (1)  0  =  S0°  and  d0  =  30' ;  (2)  r  =  2  and  d0  =  1°.  Make  a 
figure  showing  these  differentials. 

Ex.  7.  In  the  cardioid  r  =  4(1  —  cos  0)  find  the  ^-derivative  of  the  area. 
Calculate  the  differential  of  the  area  when  :  (1)  ^=60°  and  d^=l° ;  (2)  ^=0  and 
d^  =  2° ;  (3)  0  =  330°  and  d0  =  1".    Make  a  figure  showing  these  differentials. 


102  INFINITESIMAL   CALCULUS.  [Cii.  V. 

(c)  Derivative  and  differential  of  the  length  of  a  curve :  rectangular 
coordinates.     Let  PQ  be  an  arc  of  the 

curve  y=j\x).  On  FQ  take  any  point 
31{x,  y),  and  take  the  point  N{x  +  Ax, 
y  +  A?/) ;  and  draw  the  chord  MN.  On 
denoting  the  length  of  the  arc  PM  by  s, 
the  length  of  the  arc  MN  may  be  denoted 
by  As. 

Now  it  follows  from  Art.  19,  Ex.  6,  Note, 
and  the  theory  of  limits   (Arts.    20,    21), 


O 


Fig.  26.  that 

T  arc  JfiV"     !;_  chord  ilfJV  .^^ 

lnnAx=o =limA*=o :  (1) 

Ax  Ax 

lnnAx=o  ~—  =  lnnAx=o  — ^ — -—■ —  =  limAi=o  \'  1  +     .     |  • 

Ax  A^  X        VAX/ 


^=Vi+(^)-^-  (^> 


doc      ^        \dx 


From  (2),  (3),  and  Art.  27,        

and  ds  =  ^ll+{^^y.dv.  (5) 

Ex.  8.   Show  that  for  a  given  dx  and  the  actual  derivative  -  -  at  M,  the 

second  member  of  (4)  gives  the  length  of  the  intercept  of  the  tangent, 
namely,  MT.  Show  that  for  a  given  dx,  and  using  dy  to  denote  the  exact 
corresponding  change  in  the  ordinate,  the  second  members  in  (4)  and  (5) 
give  the  length  of  the  chord  of  the  arc,  namely,  the  line  MN. 

Note.  It  is  shown  in  Art.  137  how  to  find  the  length  of  the  arc  MN 
corresponding  to  an  increment  dx  in  x.  The  smaller  dx  is,  the  more  nearly 
will  MT,  arc  MN,  and  chord  3IN,  become  equal  to  one  another.  See  Ex.  6, 
Art.  19. 

Ex.  9.  (1)  Calculate  the  x-derivative  and  the  ?/-derivative  of  the  arc 
of  the  parabola  ?/2  =  4  ax.     (2)   Find  the  x-derivative  of  the  hypocycloid 

x^  +  y^  —  a^. 

Ex.  10.  In  the  cubical  parabola  y  =  x^  calculate  the  differential  of  the  arc 
at  the  point  (2,  8)  when  :  (1)  dx  =  .2  \  (2)  dy  =  A.  Show  these  differen- 
tials in  a  figure.  (The  actual  increments  of  the  arcs  can  be  computed  by 
Art.  137.) 


67.] 


GEOMETRIC  DERIVATIVES. 


103 


(d)    Derivative  and  diflFerential   of  tlie 
lengtli    of   a    curve:    polar    coordinates. 

Let  FQ  be  an  arc  of  the  curve  /(r,  0)  =  0, 
On  PQ  take  any  point  V(r,  6),  and  take 
ir(r  +  Ar,  e  +  Ad).  Denote  the  length  of  PF 
by  s  ;  then  the  length  of  VW  may  be  denoted 
by  As.  Draw  the  chord  VW. 
Now,  as  in  (c), 

chord  VW 


lim 


Ae=o' 


arc  VW 

Ad 


li.e.  ^) 
V        ddl 


imA^^o 


Ad 


(1) 


Fig.  27. 


About  0  describe  a  circular  arc  VM  intersecting  OW  in  M,  and  draw  VT 
at  right  angles  to  0  W.     Then  angle  VO  W  =  Ad,  and  M  W  =  Ar. 

.-.  TW  =  OW  -  OT  =  r  -h  Ar  -  r  cos  Ad,  and  VT  =  r  sin  A^. 


/.  chord  VW  =  V(  VT)'^  +  (  rir)^  =  V(r  sin  A^)^  +  [r(l  -  cos  A^)  +  Ary. 
chord  Ffr 


A^ 


A^    y    ■  |_         ^A^ 
chord  VW 


r 2 —  .  sill  k  Ad  -\ , 

^  A^J 


Arn2^ 


(2) 


A^ 


V-^CD' 


since,  if  A^  -  0,    5H^  -  1,  ^J^^^  =  1,  and  sin  i  A^  =  0. 
'       Ai/  ^A^  ^ 


Hence,  by  (1), 


ds 

de 


V-^f)' 


(3) 


A^ 


On  multiplying  each  member  of  (2)  by  — ,  and  then  letting  A^,  and  con- 

Ar 


sequently  Ar,  approach  zero,  it  will  be  found  that 


dr      '      \drj 
From  (3),  (4),  and  definition  Art.  27, 


and 


(4) 

(5) 
(6) 


Ex.  11.  Find  the  derivative  of  the  arc  of  the  spiral  of  Archimedes  r  =  ad : 
(1)  with  respect  to  the  angle  ;  (2)  with  respect  to  the  radius  vector. 

Ex,  12.  Calculate  the  differential  of  the  arc  of  the  Archimedean  spiral 
r  =  2d  when  d  =  2  radians  and  dd  =  1°.  Make  a  figure,  (The  actual  incre- 
ment of  the  arc  can  be  computed  by  Art.  138.) 


104 


INFINITESIMA L   CALCUL  tlS. 


[Ch.  V. 


(e)  Derivative  and  diiferential  of  the  volume  of  a  surface  of  revolu- 
tion.     Let  FQ  be  an  arc  of  the  curve  y  =f{x).      On  PQ  take  any  point 

L{x^  y),  and  take  the  point  M(x  -\-  Ax, 
y  +  Ay).  On  letting  V  denote  the  volume 
obtained  by  revolving  arc  PL  about  OX, 
the  volume  obtained  by  revolving  arc  LM 
may  be  denoted  by  A  F.  Through  L  and 
M  draw  the  lines  shown  in  the  figure. 

The  volume  obtained  by  revolving  arc  LM 
about  the  x-axis  is  greater  than  the  volume 
obtained  by  revolving  LG^  and  is  less  than  the 
volume  obtained  by  revolving  KM.    That  is, 

ir.lJL^  .LG<  AF<7r  .  Vm'^  .  KM ; 


Try^  '  Ax  <:  AV <:  w  •  (if  +  AyY  •  Ax. 
Ax 


(1) 


On  letting  Ax  approach  zero,  the  three  numbers  in  (1)  become 

7r?/2,  -— ,  7r!/2,  respectively. 
dx 


Hence, 


dV 
doc 


=  iry'K 


(2) 


From  (2)  and  Art.  27 


dV=  ir?/2  .  dx. 


If  PQ  had  been  revolved  about  the  y-axis,  then 
dV 


dy 


=  irx^f  and  dV  =  irx^'dy. 


(3) 


(4) 


Note.  According  to  (3),  for  a  given  differential  dx  the  corresponding 
differential  of  the  volume  is  the  volume  of  a  cylinder  of  radius  y  and  height 
dx.  The  smaller  dx  is,  the  more  nearly  does  this  volume  become  equal  to  the 
actual  increment,  due  to  dx,  in  the  volume  of  the  solid  of  revolution. 

Ex.  13.   Derive  the  results  in  (4). 

Ex.  14.  (1)  Find  the  x-derivative  of  the  volume  generated  by  the  revolu- 
tion of  the  parabola  y  =  x^  about  the  x-axis.  (2)  Find  the  ^/-derivative  of 
the  volume  generated  by  the  revolution  of  this  curve  about  the  y-axis. 

Ex.  15.  (1)  Calculate  the  differential  of  the  volume  in  Ex.  14  (1),  taking 
dx  =  .l  at  the  point  where  x  =  2.  (2)  Thus  also  in  Ex.  14  (2),  taking 
dy  =  .2  at  the  point  where  x  =  4.  (The  actual  increment  in  the  volume  of 
the  solid  due  to  changes  dx  and  dy  can  be  computed  by  Art.  112.) 


67.] 


GEOMETRIC  DERIVATIVES. 


105 


Y 


O 


(/)  Derivative  and  differential  of  the  area  of  a  surface  of  revolu- 
tion. Let  FQ  be  an  arc  of  the  curve  y  =f{x).  On  PQ  take  any  point,  say 
Z(x,  y),  and  take  the  point  M{x  +  Ax,  y  +  Ay).  Let  S  denote  the  area  of 
the  surface  generated  by  revolving  arc  PL  about  OX ;  then  the  area  generated 
by  revolving  arc  LM  about  OX  may  be  de- 
noted by  A.S.  There  is  evidently  a  straight 
line  whose  length  is  equal  to  the  length  of  the 
arc  LM.  Through  L  and  31  draw  the  lines 
LM'  and  ML'  parallel  to  OX  and  equal  in 
length  to  the  arc  LM.  {LM  may  be  supposed 
to  be  a  piece  of  wire,  LM'  the  same  piece  of 
wire  when  it  is  stretched  out  in  a  horizontal 
straight  line  from  Z,  and  ML'  the  same  piece 
of  wire  when  it  is  stretched  out  in  a  horizontal 
line  from  M. )  The  surface  obtained  by  revolving  the  arc  LM  about  OX  is 
greater  than  the  surface  obtained  by  revolving  LM' ;  for,  with  the  exception 
of  the  point  Z,  each  point  on  LM  has  a  greater  ordinate  than  the  corre- 
sponding point  in  the  line  LM^  and  consequently  a  greater  radius  of  swing. 
Similarly,  the  surface  obtained  by  revolving  LM  \&  less  than  the  surface 
obtained  by  revolving  ML'.     That  is, 

2  Try  .  LM'  <  surface  generated  by  L3K.2  ir  (y  +  Ay)  •  L'M; 

i.e.  2  Try  •  arc  Z3/<  A/S"  <  2  tt  (y  +  Ay)  •  arc  LM.  (1) 

(2) 


Fig.  29. 


...2.y^^^<M<2.(y  +  Ay)^I^. 
Ax  Ax  Ax 


On  letting  Ax  approach  zero,  the  three  numbers  in  (2),  by  Arts.  20,  22, 
23,  67c,  take  the  values 


and  hence 


2  Try—,   — ,   2  Try—,    respectively; 
dx    dx  dx 

^=2Try^. 
dx  dx 


On  dividing  the  members  in  (1)  by  Ay,  and  letting  Ay  approach  zero. 


^  =  2Try^. 
dy  dy 


Similarly,  if  arc  PQ  revolve  about  the  y-axis 

^  =  2Trx^     (5),     and- 
dx  dx 

From  (3),  (4),  and  Art.  67  (c)  [(2),  (3)] 


dS  o  (is  ,rN  „„j  ^dS  o  ~^s 
—  =2Trx —  (5),  and*" —  =  2Trx — • 
dx  dx  dy  dy 


(8) 


(4) 


(6) 


f=-W'^(ir-f=-w^-(ir 


106  INFINITESIMAL   CALCULUS.  [Ch.  V. 

Similarly,  in  case  of  revolution  about  the  ?/-axis,  from  (5)  and  (6), 


dx 


Results  (3),  (4),  (7),  show  that,  for  a  curve  revolving  about  the  a?-axis, 
dS  =  2'!^yds  =  2  irj/^l  +  {^^£fdoc  =  2  ^ijyjl  +(^Ydij ;      (9) 


and  (5),  (6),  (8),  show  that,  for  a  curve  revolving  about  the  2/-axis 


dS  =  2'irx*ds  =  2  irxyjl  +  I^^Y€lsc  =  2  irscyjl  +  l^Y  dy.       (10) 

Ex.  16.    Derive  results  (5),  (6),  (8),  and  (10). 

Ex.  17.  Find  the  x-derivative  and  the  ^/-derivative  of  each  of  the  surfaces 
described  in  Ex.  14. 

Ex.  18.  Calculate  the  differentials  of  the  surfaces  described  in  Ex.  15. 
Make  figures  showing  these  differentials.  (The  actual  increments  of  the 
surfaces  can  be  computed  by  Art.  139.) 

Ex.19.    Find    ^,    ^^— ,   ^,   ^,  for  the  ellipse  h'^x^  +  a^y'^  =  a%-^.     For 
dx     dx     dx     dx 

a  given  differential  of  x,  draw  figures  showing  the  corresponding  differentials 
of  s,  A^  F,  and  x. 

d<i 

Ex.20.  Find  —  for  r^=a^cos2d,  r=acosd,  r=ae»«=ot«,  r=:a(l+cos^). 
dd 

Ex.  21.    If  0  denote  the  eccentric  angle  of  the  ellipse  in  Ex.  19,  show  that 

—  =  ay/l  —  e^  cos"^  0,  e  being  the  eccentricity. 
d(p 


CHAPTER   VI. 

SUCCESSIVE  DIFFERENTIATION. 

N.B.  Article  68  contains  all  that  the  beginner  will  find  necessary  concern- 
ing successive  differentiation  for  the  larger  part  of  the  remaining  chapters. 
Accordingly,  the  reading  of  Arts.  69-72  may  be  deferred  until  later. 

68.  Successive  derivatives.  As  observed  in  many  of  the  pre- 
ceding examples,  the  derivative  of  a  function  of  x  is,  in  general, 
also  a  function  of  x.  This  derivative,  which  may  be  called  the 
Jirst  derived  function,  or  the  first  derivative  (of  the  function),  may 
itself  be  differentiated ;  the  result  is  accordingly  called  the  second 
derived  function,  or  the  second  derivative  (of  the  original  function). 
If  the  second  derivative  is  differentiated,  the  result  is  called  the 
third  derived  function,  or  the  third  derivative  ;  and  so  on.  If  the 
operation  of  differentiation  is  performed  on  a  function  n  times  in 
succession,  the  final  result  is  called  the  ?ith  derived  function,  or 
the  nth.  derivative,  of  the  function. 

Ex.  If  the  function  is  x*,  then  its  first  derivative  is  Ax^  \  its  second 
derivative  is  12  x"^;  its  third  derivative  is  24  x;  its  fourth  derivative  is  24; 
its  fifth  and  its  succeeding  derivatives  are  all  zero. 

dotation.     («)  If  y  denote  the  function  of  x,  then 

the  first  derivative,  namely  — (jj),  is  denoted  by  ~^   (Art.  23) ; 

cix  ax 

the  second  derivative,  namely  — (  —  |,  is  denoted  by  J; 

dx\dxj  dx^ 


the  third  derivative,  namely  — 

dx 


"Af^^],  is  denoted  by  ^: 
dx\dx)]  ^  d^' 


and  so  on.     On  this  plan  of  writing, 


the  nXk  derivative  is  denoted  by  — ^. 

107 


108  INFINITESIMAL   CALCULUS.  [Ch.  VI. 

In  this  notation  the  integers  2,  3,  •••,  n,  are  not  exponents; 
these  integers  merely  indicate  the  number  of  times  that  the  func- 
tion y  is  to  be  differentiated  successively  with  respect  to  x. 

(6)  The  letter  D  is  frequently  used  to  denote  both  the  opera- 
tion and  the  result  of  the  operation  indicated  by  the  symbol 

d 
— -     (See  Art.  23.)     The  successive  derivatives  of  y  are  then 

(XX 

Dy,  D{Dy),  I>lD(Dy)],  ••• ;  these  are  respectively  denoted  by 

Dy,Dhj,I)'y,^.;D-y. 

Sometimes  there  is  an  indication  of  the  variable  with  respect 
to  which  differentiation  is  performed ;  thus 

D^y,DJy,D^%^.',Dj^y. 

Note.  Here  n  is  not  an  exponent ;  D"?/  does  not  mean  {Dyy\  {E.g.  see 
Ex.,  p.  107.)     i)"y  is  called  the  derivative  of  the  nth  order. 

(c)  Instead  of  the  symbols  shown  in  (a)  and  (6),  for  the  succes- 
sive derivatives  of  y,  the  following  are  sometimes  used,  namely, 

(d)  If  the  function  be  denoted  by  <fi(x),  its  first,  second,  third,  •••, 
and  7ith.  derivatives  (with  respect  to  x)  are  generally  denoted  by 

<f>'{x),  <f>"{x),  <fi"'(x),  •••,  (^("^(a?)  or  </>"(cc),  respectively. 

Note  1.  In  this  book  notation  (a)  is  most  frequently  used.  The  symbol 
Z>  is  very  convenient,  and  is  especially  useful  in  certain  investigations.  See 
Byerly's  Diff.  CaL,  Lamb's  Calculus,  Gibson's  Calculus  (in  particular  §  67). 
For  an  exposition  of  simple  elementary  properties  of  the  symbol  D  also  see 
Murray's  Differential  Equations  (edition  1898),  Note  K,  page  208. 

Note  2.     Instead  of  the  accent  notation  in  (c),  the  'dot '-age  notation, 

y,  y\  'y\  ••• 

is  sometimes  used,  particularly  in  physics  and  mechanics. 

Note  3.    Geometrical  meaning  of  ^^«    It  has  been  seen  in  Arts.  25,  26, 
du  d  dx^ 

that  -^,  i.e.  —  (?/),  denotes  the  rate  of  change  of  y,  the  ordinate  of  the  curve, 

dx  dx 

compared  with  the  rate  of  change  of  the  abscissa  x ;  this  may  be  simply 

denoted  as  the  a;-rate  of  change  of  the  ordinate.    Similarly  — ^,  i.e.  —  I  —  ). 

dx^         dx  \dxj^ 

is  the  rate  of  change  of  the  slope  -^  of  a  curve  compared  with  the  rate  of 

dx 
change  of  the  abscissa  x,  or,  simply,  the  aerate  of  change  of  the  slope. 


68.]  SUCCESSIVE  DIFFERENTIATION.  109 

On  a  straight  line,  for  instance,  the  slope  is  constant,  and  hence  the  x  rate 
of  change  of  the  slope  is  zero.     This  is  also  apparent  analytically.     For,  if 

y  =  mx  +  c  is  the  equation  of  the  line,  then  -^  =  m,  and  hence  -^  =  0. 

dx  dx^ 

Note  4.    Physical  meaning  of  ^^»     In  Art.  25  it  has  been  seen  that 

ds  d 

if  s  denotes  a  varying  distance  along  a  straight  line,  — ,  i.e.  —  (s),  denotes 

Clt  civ 

the  rate  of  change  of  this  distance,  i.e.  a  velocity.    Similarly  —,  i.e.  —  (  —  V 

dt^         dt  \dt ) 

denotes  the  rate  of  change  of  this  velocity.     Rate  of  change  of  velocity  is 

called  acceleration.    For  instance,  if  a  train  is  going  at  the  rate  of  30  miles 

an  hour,  and  half  an  hour  later  is  going  at  the  rate  of  40  miles  an  hour,  its 

velocity  has  increased  by  '  10  miles  an  hour'  in  half  an  hour,  i.e.  as  usually 

expressed,  its  acceleration  is  10  miles  per  hour  per  half  an  hour.     Again,  it 

is  known  that  if  s  is  the  distance  through  which  a  body  falls  from  rest 

in  t  seconds,  s  =  |  gt^.     Hence  —=  gt;  accordingly,  -^=  g.    That  is,  the 

dt  dt" 

acceleration  of  a  falling  body  is  '^  feet  per  second'  per  second.  (See 
text-books  on  Kinematics,  Dynamics,  and  Mechanics,  for  a  discussion  on 
acceleration.) 

EXAMPLES. 

1.  Find   the    second    a:-derivative    of:     (i)  xtan-ix;     (ii)  'ix^  —  9x  + 

o  _ 

Vx  ;  (iii)  tan  x  +  sec  x  ;    (iv)  x'. 

X 

2.  Find  D/y,  when  :  (i)  y=  {x?  +  a^)  tan"!  -  ;    (ii)  y  =  log  (sin  x). 

a 

3.  Find  — ^,  when  :  (i)  y  =  sin-i  x  \    (ii)  y  = 

dx^  1  +  x^ 


5.    Find  ^,  when  a!?/2  +  3  a:  +  5  y  =  0. 

By  Art.  56,  ^  =  -   ^^  +  ^  ■  (1) 


4.    Find  D^y,  when  :  (i)  y  =  a;*  log  x  ;  (ii)  y  =  e^  cos  x 

cPy 
dx^ 

dx         2xy  +  b 

On  differentiation,       ^^  = ^ ^ ^• 

dx2  (2  xy  +  5)2 

On  substituting  the  value  of  -^,  and  reducing, 
dx 

d^ ^  2(y2  +  3)  (3 xy'^  +  lOy-Sx)  .gv 

dx^  (2  xy  +  5)3  ^  ^ 

6.    (i)   In    the    ellipse    a^y^  +  hH^  =  a^lP'    calculate    B^y.       (ii)    Given 
y2  +  y  =  x2,  find  B^. 


110  INFINITESIMA£    CALCULUS.  [Ch.  VI. 

7.  Show  that  the   point   (J,  ^)   is   on   the   curve   log  (x  +  y)  =  x  —  y. 

Show  that  at  this  point  -^  =  0,  and  ^'-^-  =  i. 
(k:  dx'^ 

8.  What  are  the  values  of  -^  and   --^^ :     (i)  at   the   point  (2,  1)   on 

the   ellipse    7  ic^  +  10  ?/2  =  g8  ;    (ii)    at  the   point    (3,    5)    on  the   parabola 
y2  —  4x  -\-  13. 

9.  Calculate  ^  for  the  cycloid  in  Art.  43,  Ex.  6.     Compute  it  when 

3' 

10.  Verify  the  following  :  (i)  if  y  =  as[nx-{-  b  cos  x,  ^  -{- y  =  0  ; 
(ii)  if  u  =  (sin-i  xy,  (1  -  a:2)  i^  -  a:  ^^  =  2  ;  (iii)  \i  y  =  a  cos  (log  x)  + 
h  sin  (log  X) ,  a;2  ^^  +  x  ^-'^  +  ?/  =  0. 


11.    Show  tlmt   if   11 
y(21og2/  +  l)|^. 


2/2 log 2/,    and  2/=/(x),    l?^=(21og?/  +  3)f^V 


12.  Find  ^  in  the  following  cases :  y  —  A:X^  ^1x-Z,  y  =  4iK3  + 
4  x  +  2,  y  =  4  x3  +  5  a;  —  4,  2/  =  4  x^  +  ex  +  ^•. 

13.  Given   that     _  -^  =  3  x  +  2,    find    the    most    general    expression   for 

^  OjX 

-^  ;  then  find  the  most  general  expression  for  y. 
ax 

14.  A  curve  passes  through  the  point  (2,  3)  and  Its  slope  there  is  1  ; 

at  any  point  on  this  curve    -^  =  2  x  :  find  its  equation  and  sketch  the  curve. 
d'y? 

15.  At  any  point  on  a  certain  curve   -^  =  8 ;  the  curve  passes  through 

dx^ 
the  origin  and  touches  the  line  y  =  x  there  ;  find  its  equation  and  sketch  the 
curve. 

16.  (1)  In  the  case  of  simple  harmonic  motion,  Ex.  13  (p.  83),  show 
that  the  speed  of  ^  is  changing  at  a  rate  which  varies  as  the  distance  of  ^ 
from  the  centre  of  the  circle.  (2)  What  is  the  acceleration  of  the  velocity 
of  the  boat  in  Ex.  18,  Art.  37  ? 

17.  In  Ex.  14  (p.  83),  calculate  the  rate  at  which  Q_  is  olianging  its 
speed  when  ^  is  :  (i)  at  an  extremity  of  the  diameter  ;  (ii)  12  inches  from 
the  centre  ;    (iii)  6  inches  from  the  centre  ;    (iv)  at  the  centre. 

18.  A  body  moving  vertically  has  an  acceleration  or  a  retardation  of 
g  feet  per  second  due  to  gravitation,  g  being  a  number  whose  approximate 
value  is  32.2  :  find  the  most  general  expression  for  the  distance  of  tli«  moving 
point  from  a  fixed  point  in  its  line  of  motion,  after  t  seconds.  Explain  the 
physical  meaning  of  the  constants  that  are  introduced  in  the  course  of 
integration. 


69.]  SUCCESSIVE  DIFFERENTIATION.  Ill 

19.  A  body  is  projected  vertically  upwards  with  an  initial  velocity  of 
500  feet  per  second  :  liiid  how  long  it  will  continue  to  rise,  and  what  height 
it  will  reach,  if  the  resistance  of  the  air  be  not  taken  into  account. 

20.  A  rifle  ball  is  fired  through  a  three-inch  plank,  the  resistance  of 
which  causes  an  unknown  constant  retardation  of  its  velocity.  Its  velocity 
on  entering  the  plank  is  1000  feet  a  second,  and  on  leaving  the  plank  is 
500  feet  a  second.  How  long  does  it  take  the  ball  to  traverse  the  plank  ? 
(Byerlj'^,  Problems  in  Differential  Calculus.) 

69.   The  /7th  derivative  of  some  particular  functions.     In  a  few 

cases  the  nt\\  derivative  of  a  function  can  be  found.  This  is 
done  by  differentiating  the  function  a  few  times  in  succession, 
and  thereby  being  led  to  see  a  connection  between  the  successive 
derivatives. 

EXAMPLES. 

1.  Let  y  =  cc'". 
Then                              Dy  =  rx''-^ ; 

j)2y  =  r(^r  -  l)a;'-2  ; 

j)3y  ^  r(r  -  l)(r  -  2)x'-3. 
From  this  it  is  evident  that 

D»y  =  r(r  -  1)  (r  -  2)  •••  (r  -n  +  l)x'-«. 
Show  that  i>»a;»»  =  w  ! 

2.  Find  the  nth  derivative  of  the  following  functions  : 

(a)  e*;  (6)  a^ ;  (c)  e«*;  (d)  aK 

3.  Show  that  the  wth  derivative  of  sinx  is  sin   (x  +  — V 
Suggestion:    coss  =  sin  f  2r  +  ^ V 

4.  Find  the  wth  derivatives  of  (a)  cos  x  ;    (&)  sin  ax  ;    (c)  cos  ax. 

5.  Find  the  ?ith  derivatives  of  log  x,  log  (x  —  2)2. 

6.  Find  the   wth    derivatives    of    -,       ^  ^  ^ 


X     I  -\-x    3-x     (6  +  ra)'" 


7.    Find   the    nth    derivatives  of    — — ,    -^ 

1  -  a:^     1  -  a 

[Suggestion  :   Take  the  partial  fractions.] 


112  INFINITESIMAL   CALCULUS.  [Ch.  VI. 

70.  Successive  differentials.     In  Art.  27  it  has  been  shown  that  if 

y=f(x),  (1) 

then  dy=f'(x)dx.  (2) 

The  differential  in  (2)  is,  in  general,  also  a  function  of  x ;  and  its  differ- 
ential may  be  required.  In  obtaining  successive  differentials  it  is  usual  to 
give  a  constant  differential  increment  dx  to  x.  Then  (Art.  27),  on  taking 
the  differentials  of  the  members  in  (2), 

didy)  =  d  lff(x)dx^  =  [f"(x)dx']  dx.  (3) 

On  taking  the  differentials  of  the  members  of  (3), 

d{didy)}  =  d{[f"(x)dx']dx}=f>"(x)dx  •  dx  -  dx.  (4) 

It  is  customary  to  denote  results  (o)  and  (4)  thus  : 

d^y=f"ix)dx-^  and  d^y  =f"'(x)dxK 

In  this  notation  the  nth  diflPereatial  is  written 

in  which  /"(x)  denotes  the  nth  derivative  of  f(x),  and  dx»  denotes  (dx)». 

71.  The  successive  derivatives  of  /  with  respect  to  x  when  both 
are  functions  of  a  third  variable,  /  say. 

By  Art.  26,  Note  1,  ^  =  ~ 

dt 

'  ^  —  A.  f^^  —  ^  f^l\  .  ^     C^y  ^^®  principle  in  Art.  34, 
dx^~~  dx\dx)  ~  dt\dxj  '  dx        Eq.  (2)] 

dx  ^ _dy  ^  ^  dx    d^y     dy    d^x 

dt  '  df      dt  '  df     dt      dt  '  df      dt  '  df 


fdxV 


dx  fdx^ ^ 

dt 


The   method   of   obtaining  the  higher  derivatives  is  similar. 

Thus,       d^^d_/^\^df^\  ^  d£^d/(Py\^dx^ 
dx^     dx\dxy      dt\dx'J  '  dx     dt\dxy  '  dt' 


70-72. J  SUCCESSIVE  DIFFEEENTIATION.  113 

And,  in  general, 

dx"  ~  dx  [dx""-^)  ~  dt  \dx''-^)  '  dx~  dt  \dx''-y 


dx 
~dt 


Ex.  1.    See  Ex.  9,  Art.  68. 

Ex.  2.    Find  D^y  and  D^y  when  x  =  a  —  b  cos  6  and  y  =  ad  +  bsin  0. 

72.  Leibnitz's  theorem.  This  theorem  gives  a  formula  for  the  nth  deriva- 
tive of  the  product  of  two  variables.  Suppose  that  u  and  v  are  functions  of 
X,  and  put  y  =  uv. 

Then,  on  performing  successive  differentiations, 

Dy  =  u  •  Dv  -\-  V  '  Du  ; 

D^y  =  u  .  DH  +  2  Du-  Dv  -\-v  •  DHi ; 

Dhf  =  u-DH^^  Du  .  D^v  +  3  DHi  •  Dv -^  v  -  Dhi ; 

D^y  =  u  '  Dh  -{-  4  Du  ■  DH  -\-  6  Dhi  ■  D'v  +  4  Dhi  -  Dv  +  v  -  D^u. 

Thus  far  the  numerical  coefficients  in  these  derivatives  are  the  same  as  the 
numerical  coefficients  in  the  expansions  (m  +  v),  (m  +  i?)^,  (u  +  v)^,  and 
(u  +  vy  respectively,  and  the  orders  of  the  derivatives  of  u  and  v  are  the 
same  as  the  exponents  of  u  and  v  in  those  binomial  expansions.  Now  sup- 
pose that  these  laws  (for  the  numerical  coefficients  and  the  orders)  hold  in 
the  case  of  the  ?ith  derivative  of  uv ;  that  is,  suppose  that 

D'^{nv)  =  u  '  D^v  +  nDu  •  D'^-^v  +  ^^^  ~  ^^  DH  •  D^-^v  +  ... 

n(n-l).-(n-r  +  2)  j^,_^^  _  ^„_,^i^ 
1.2...(r-l) 

+  ^'^  -  ^>-<''  -''^^^Dru.  D-ry  +...  +  !;.  Pr^u.     (1) 
1  .  2  •••  r 

Then  these  laws  for  the  coefficients  and  the  orders  hold  in  the  case  of  the 
(w  +  l)th  derivative  of  uv.     For  differentiation  of  both  members  of  (1)  gives 

Z)"+i(wv)  =  u  .  D^+iv  +  (w  +  V)Du  •  D^v  +  i'^'^^)''^  Dhi  •  D'^-^v  +  ••• 

1  •  ^ 

+  {n  +  l)n(n-l).--(n-r  +  2)  ^,^  ^  j^n-r+i^  +  ...  +  v  D-+^u. 
l-2...(r-l)r 

Hence,  if  formula  (1)  is  true  for  the  nth  derivative  of  uv,  a  similar  formula 
holds  for  the  (w  +  l)th  derivative.  But,  as  shown  above,  formula  (1)  is  true 
for  the  first,  second,  third,  and  fourth  derivatives  of  uv  ;  hence  it  is  true  for 
the  fifth,  and  for  each  succeeding  derivative. 


114  INFINITESIMAL   CALCULUS.  [Ch.  VI. 

Ex.  1.    Find  2)/'«/  when  ij  =  aj^e^. 

=  e*[x2  +  2  nx  +  ?i(«  -  1)]. 

Ex.  2.    Calculate  the  fourth  :*;-derivative  of  x^  sin  x  by  Leibnitz's  theorem. 

Ex.  3.    Find  Dx'^y  when  :  (i)  y  =  xt""  \  (ii)  y  —  a'e-^. 

Note.  Reference  for  collateral  reading  on  successive  differentiation. 
Echols,  Calculus,  Chap.  IV.,  especially  Art.  56. 

73.  Application  of  differentiation  to  elimination.  It  is  shown  in 
algebra  that  one  quantity  can  be  eliminated  between  two  inde- 
pendent equations,  two  quantities  between  three  equations,  and 
that  n  quantities  can  be  eliminated  between  n-|-l  independent 
equations.  The  process  of  differentiation  can  be  applied  for  the 
elimination  of  arbitrary  constants  from  a  relation  involving  vari- 
ables and  the  constants.  For  by  differentiation  a  sufficient  num- 
ber of  equations  can  be  obtained  between  which  and  the  original 
equation  the  constants  can  be  eliminated. 

EXAMPLES. 

1.    Given  that  y  =  Acosx  -{-  B sin  x,  (1) 

in  which  A  and  B  are  arbitrary  constants,  eliminate  A  and  B. 

In  order  to  render  possible  the  elimination  of  these  two  constants,  two 
more  equations  are  required.  These  equations  can  be  obtained  by  differen- 
tiation.    Thus, 

^  =  -Aiimx  +  Bcosx,  (2) 

dx 

-r-^  =  —  Acosx  —  B sin  x.  i^) 

dx^ 

On  eliminating  A  and  B  between  (1),  (2),  (3),  there  is  obtained  the  relation 

Note  1.  Equation  (4)  is  called  a  differential  equation,  as  it  involves  a 
derivative.  It  is  the  differential  equation  corresponding  to,  or  expressing 
the  same  relation  as,  the  "integral"  equation  (1).  The  process  of  deducing 
the  integral  equations  (or  solutions,  as  they  are  then  called)  of  differential 
equations  is  discussed,  but  for  a  very  few  cases  only,  in  Chapter  XXI. 


73.]  SUCCESSIVE  DIFFERENTIATION.  115 

2.  Eliminate  the  arbitraiy  constants  m  and  b  from  the  equation 

y  =  mx  +  b.  Ans.  -^  =  0. 

In  this  case  the  given  equation  represents  all  lines,  m  and  b  being  arbi- 
trary. Accordingly  the  resulting  equation  is  the  differential  equation  of  all 
lines.     For  the  geometrical  point  of  view  see  Art.  68,  Note  3. 

3.  Eliminate  the  arbitrary  constants  a  and  b  from  each  of  the  follovr- 
ing  equations  :   (1)  ?/  =  wx^  -f  6.       (2)  y  =  ax^  +  bx.      (3)   (j/  —  6)2  =  4  ax. 

(4)  y-^  -  2  a?/  +  x2  =  a^.      (5)  y^  =  b{aP-  -  x^). 

4.  Find  the  differential  equations  which  have  the  following  equations 
for  solutions,  Ci  and  Cg  being  arbitrary  constants  : 

(l)y  =  ci.  {2)y  =  cix.  (3)  ?/ =  cix  +  C2.  (4)  y  =  Cie*  +  026"=^. 

(5)  y  =  cie'^  +  C2e~™'=.    (6)  ?/  =  Ci  cos  mx  +  C2  sin  wx.    ( 7 )  ?/  =  ri  cos  (mx  +  C2) . 

5.  Obtain  the  differential  equations  of  all  circles  of  radius  r:  (1)  which 
have  their  centres  on  the  x-axis ;  (2)  which  have  their  centres  on  the  y-axis ; 
(3)  which  have  their  centres  anywhere  in  the  xy-plane. 

6.  Show  that  the  elimination  of  n  arbitrary  constants  Ci,  C2,  •••,  c„, 
from  an  equation  f(x,  ?/,  Ci,  C2,  •••,  c„)  =  0  gives  rise  to  a  differential  equation 
involving  the  nth  derivative  of  y  with  respect  to  x. 

Note  2.  For  geometrical  explanations  relating  to  differential  equations 
the  student  is  referred  to  Murray,  Differential  Equations.,  Chap.  I.,  wliich  may 
easily  be  read  now.    The  reading  will  widen  his  mathematical  outlook  at  this 


CHAPTER   VII. 

FURTHER   ANALYTICAL   AND    GEOMETRICAL 
APPLICATIONS. 

VARIATION  OF   FUNCTIONS.      SKETCHING  OF   GRAPHS. 
MAXIMA   AND    MINIMA.      POINTS   OF   INFLEXION. 

N.B.  This  chapter  may  be  studied  before  Chap.  V.  is  entered 
upon. 

74.  Increasing  and  decreasing  functions.  When  x  changes  con- 
tinuously from  one  value  to  another,  any  continuous  function  of  x, 
say  <f>(x),  in  general  also  changes.  The  function  may  either  be 
increasing  or  decreasing,  or  alternately  increasing  and  decreas- 
ing. By  means  of  the  calculus  it  is  easy  to  find  out  how  the 
function  behaves  when  x  passes  through  any  value  on  its  way 
from  —  00  to  -f-  GO. 

Let  Ax  be  a  positive  increment  of  x,  and  A<f>(^x)  be  the  corre- 
sponding increment  of  cf)(x).  If  (f>(x)  continually  increases  when  x 
is  changing  from  a;  to  x  -|-  Ax,  then  A<f>(x)  is  positive ;  and  accord- 
ingly,    ^^  ^  is  positive.    Moreover,  this  is  positive  for  all  positive 

iJkX 

values  of  Ax,  however  small ;  hence  lim^^^o  is  positive,  i.e. 

Ax 

<^'(x)  is  positive.     In  a  similar  way  it  can  be  shown  that  if  cf>{.v) 

continually  decreases  when  x  is  changing  from  x  to  x  -f  Ax,  then 

<^'(x)  is  negative.     These  facts  may  be  stated  thus : 

<^'(x)  is  positive  when  <^(x)  is  increasing,  and  |  j 

<^'(x)  is  negative  when  <^(x)  is  decreasing ;  and  conversely.  J 

These  facts  will  also  be  apparent  on  an  inspection  of  the  accom- 
panying graphs. 

Let  <^(x)  be  graphically  represented  by  the  curve  ABCDE, 
whose  equation  is  ./  . 

116 


74,  75.] 


MAXIMUM  AND  MINIMUM. 


117 


At  any  point  on  this  curve, 


dx 


^\x). 


By  Art.  24,  the  slope  of  the  curve  represents  the  derivative  of 
the  function.  Now  at  A,  D,  and  E,  the  slope  is  negative,  and  the 
ordinate  y  (the  function)  is  evidently  decreasing  as  x  is  passing  in 
the  positive  direction  through  the  values  of  x  at  A,  D,  and  E. 
On  the  other  hand,  at  B,  C,  and  F,  the  slope  is  positive,  and  the 
ordinate  y  is  evidently  increasing  as  x  is  passing  in  the  positive 


\ 

r1 

Y 

» 

ti 

i     i^ 

0 
h 

I. 

S 

r 

^ 

L 

\ 

r 

Li 

3fi 

N, 

0 

— 

n — » 

Y 

J 

K 

A         0 

« k ' 

X 

Fig.  30  6. 


Fig.  30  c. 


Fig.  30  a. 


direction  through  the  values  of  x  at  B,  C,  and  F.  In  Fig.  30  6 
when  X  is  increasing  from  07>i  to  OM^,  the  ordinate  ?/  is  decreas- 
ing from  LiL  to  M^M  and  the  slope  at  points  on  LM  is  negative ; 
when  X  is  increasing  from  OMi  to  OiYi,  the  ordinate  is  increasing 
from  M^M  to  N^N  and  the  slope  at  points  on  MN  is  positive. 
Fig.  30  c  also  exemplifies  principles  marked  A  on  page  116. 

75.  Maximum  and  minimum  values  of  a  function.  Critical  points 
on  the  graph,  and  critical  values  of  the  variable.  The  values  of  the 
function  at  points  such  as  Pj,  P^,  P3,  M,  and  K  (Art.  74),  where 
the  function  stops  increasing  and  begins  to  decrease,  or  vice  versa, 
may  be  called  turning  values  of  the  function.  When  a  function 
ceases  to  increase  and  begins  to  decrease,  as  at  P2,  P4,  and  K,  it  is 
said  to  have  a  maximum  value  ;  when  a  function  ceases  to  decrease 
and  begins  to  increase,  as  at  Pj,  P3,  and  M,  it  is  said  to  have  a 
minimum  value.  Therefore,  at  a  point  (on  the  graph)  where  the 
function  has  a  maximum  value  the  slope  changes  from  positive  to 
negative;  at  a  point  where  the  function  has  a  minimum  value  the 
slope  changes  from  negative  to  positive.      (Examine  Fig.  30.) 


118 


INFINITESIMA L   CALCUL US. 


[Ch.  VII. 


Accordingly,  at  each  of  these  points  the  slope  (i.e.  the  derivative  of 
the  function)  is  generally  (see  j^ote  3)  either  zero  or  infinitely  great. 
It  should  be  observed  that,  although  the  derivative  of  a  function 
may  be  either  zero  or  infinitely  great  for  values  of  the  variable  for 
which  the  function  has  a  maximum  or  a  minimum  value,  yet  the 
converse  is  not  always  the  case.  The  function  may  not  have  a 
maximum  or  minimum  value  when  its  derivative  is  zero  or  infinity. 


o 
Fig.  31  b. 


This  is  exemplified  by  the  functions  whose  graphs  are  given  in 
Figs.  31  a,  b.  Thus  at  P  the  slope  is  zero  and  the  function  is 
increasing  on  each  side  of  P;  at  Q  the  slope  is  zero  and  the 
function  is  decreasing  on  each  side  of  Q;  at  i?  the  slope  is  infi- 
nitely great,  and.  the  function  is  increasing  on  each  side  of  R; 
at  S  the  slope  is  infinitely  great  and  the  function  is  decreasing 
on  each  side  of  S. 

Accordingly,  a  point  where  the  slope  of  the  graph  of  a  function 
is  zero  or  infinitely  great  is,  for  the  purpose  of  this  chapter,  called 
a  critical  point.  Such  a  jjoint  must  be  further  criticised,  or  ex- 
amined, in  order  to  determine  whether  the  ordinate  has  either  a 
maximum  or  a  minimum  value  there.  In  other  words,  that  value 
of  the  variable  for  which  the  derivative  of  a  function  is  zero  or 
infinitely  great  is  called  a  critical  value  ;  further  examination  is 
necessary  in  order  to  determine  whether  the  function  is  a  maxi- 
mum or  a  minimum  for  that  value  of  the  variable. 

Note  1.  The  points  ^,  P,  i?,  8  (Figs.  31  a,  6),  are  examples  of  what 
are  called  pom^s  of  inflexion  (see  Art.  78). 

Note  2.  By  saying  that  a  function  <f>{x)  has  a  minimum  value,  for  x  =  a 
say,  it  is  not  meant  that  0(a)  is  the  least  possible  value  the  function  can 
have.  It  is  meant  that  the  value  of  the  function  for  a;  =  a  is  less  than  the 
values  of  the  function  for  values  of  x  which  are  on  opposite  sides  of  a, 


76.] 


MAXIMUM  AND  MINIMUM. 


119 


and  as  close  as  one  pleases  to  a  ;  i.e.  h  being  taken  as  small  as  one  pleases, 
</>(a)  <  0(«  -  h)  and  </>(«)  <  ^(a  +  h).  (See  Pi  in  Fig.  30  a.)  Likewise,  if 
^(x)  is  a  maximum  for  x  =  b,  this  means  merely  that  0(6)  >  0(6  —  h)  and 
0(&)  >0(&  +  ^)»  in  which  ^  is  as  small  as  one  pleases.     (See  P2  in  Fig.  30  a.) 


EXAMPLES. 
1.    Examine  sin  x  for  critical  values  of  the  variable. 
Here  0(x)  =  sinx. 

The  graph  of  this  function  is  on  page  409.     In  order  to  find  the  critical 

0'(a;)  =  cos  a;  =  0. 


points  solve  the  equation 


Accordingly,  the  critical  values  of  x  are  — ,  — ,  — ,  •  •• . 

&j,  222' 

2.  Examine  (x  —  iy(x  +  3)  for  critical  values  of  the 
variable. 

Here  0(x)  =  (x  -  l)2(x  +  3). 

The  solution  of   0'  (x)  =  (x  -  1)  (3  x  +  5)  =  0, 
gives  the  critical  values  of  x,  viz.  1,  —  f. 

3.  Examine   (x  —1)^  +  2    for    critical  values    of    the 
variable. 

Here  0(x)  =  (x  -  1)^  +  2. 

On  solving  0'(x)  =  3(x  -  1)^  =  0, 

the  critical  value  of  x  is  obtained,  viz.  x  =  1. 

2 

4.  Examine  (x  -  2)  '^  +  3  for  critical  values  of  x. 

2 

Here 


Fi 
Y 

n  c. 

J 

/ 

/^ 

i 

lO 

-\-* 

X 

On  solving 


0(x)  =  (x-2)3  +  3. 
^'(x)  = ^  =  c 


3(x-2)3 
the  critical  value  x  ==  2  is  obtained. 

6.   Examine  (x  —  2)^  +  3  for  critical  values  of  x. 


Here 
and 


0(x)  =  (x-2)3+3, 


0'(x)  = 


=  0 


Fig.  .si  d. 
Y 


H 


Fig.  31  e. 


Y 

) 

0 

^2- 

X 

3(x  -  2)^ 
gives  the  critical  value  x  =  2.  p^^   g^  , 

Note  3.     A  function  may  have  a  maximum  or  minimum  value  when  its 
derivative  changes  abruptly  ;  see  Art.  164,  Note  3,  and  Fig.  21  (c). 


120  INFINITESIMAL    CALCULUS.  [Ch.  VIL 

76.  Inspection  of  the  critical  values  of  the  variable  (or  critical 
points  of  the  graph)  for  maximum  or  minimum  values  of  the  function. 

Let  the  function  be  </>(.^')•     The  equation  of  its  graph  is  2/  =  <^(^)? 

and  the  slope  is  -^  or  ^'{x).     The  solutions  of  the  equations 

(f>'(x)  =  0   and   <l>'{x)  =  cc, 

give  the  critical  values  of  the  variable. 

Suppose  that  ABODE  (Fig.  30  a)  is  the  graph,  and  that  the 
critical  values  are  x  =  a  and  x  =  b.  There  are  three  ways  of 
testing  whether  the  critical  values  of  the  variable  will  give  maxi- 
mum or  minimum  values  of  the  function,  viz. : 

(a)  By  examining  the  function  itself  at,  and  on  each  side  of, 
the  critical  value  ; 

(b)  By  examining  the  first  derivative  on  each  side  of  the 
critical  value ; 

(c)  By  examining  the  second  derivative  (see  Art.  68)  at  the 
critical  value. 

Note  1.  It  follows  from  the  definition  of  maximiini  and  minimum  values, 
and  Note  2,  Art.  75,  that  if  <p(a)  is  a  maximum  (or  minimum)  value  of  0(,r), 
then  0(a) -f  w,  c4>(a),  v^0(a),  0^(a),  •••,  are  maximum  (or  minimum) 
values  of  0(cc)  +  w,  c4>(x),  V<p(x),  0^(x),  •••,  respectively.  Accordingly, 
the  finding  of  critical  values  of  x  for  one  of  these  functions  will  give  the 
critical  values  for  the  other  functions.  It  sometimes  happens  that  it  is  much 
easier  to  find  the  critical  values  for,  say  ^^(x),  than  for  (p(x).  In  such  a 
case  it  is  better  to  examine  (p'^(x)  than  to  examine  ^(x). 

(a)  Examination  of  the  function.  In  this  test,  if  x=^a  is  the 
critical  value,  (f>(a  —  h)  and  cf>(a-i-h),  in  which  h  is  as  small  as 
one  pleases,  are  both  compared  with  cf>{a).  (This  is  the  obvious 
and  natural  method  .of  testing  the  critical  values.)  If  <l>(ci)  is 
greater  than  both  <j>{a  —  li)  and  <f>{a-\-h),  then  <fi(a)  is  a  maxi- 
mum; if  <f>{a)  is  less  than  both  <j>{a  —  h)  and  <f>(a-\-h),  then 
<f>{a)  is  a  minimum  ;  if  <fi(a)  is  greater  than  one  and  less  than 
the  other  of  <l>(a  —  h)  and  <^(a4-/i),  then  <j>(a)  is  neither  a 
maximum  nor  a  minimum. 

Ex.  1.    In  Ex.  1,  Art.  75,  examine  the  function  at  the  critical  value  J  of  x. 

.. )  <  sin  -,  and  sin  (  -  +  /i  J  <  sin  - .     Accordingly,  a^  =  ^ 
gives  a  maximum  value  of  sin  x. 


76.]  MAXIMUM    AND    MINIMUM.  121 

Ex.  2.  (a)  In  Ex.  2,  Art.  75,  examine  the  function  at  the  critical  value 
x  =  \.  Here  0(1)  =  O,  </)(l -/i)  =  /i-2(4-/i),  0(l  +  /i)  =  /i2(4  +  /i).  Accord- 
ingly, 0(1  —  /i)  >  0(1),  and  0(1  -\-  h)>  0(1).  Thus  0(1)  is  a  minimum 
value  of  0(x). 

(6)  Inspect  this  function  at  the  critical  value  a;  =  —  |. 

Ex.  3.  In  Ex.  3,  Art.  75,  examine  the  function  at  the  critical  value  x  =  1. 
Here  0(1)  =  2,  (t>(\  -  h)  =  - h^  ■\-2,  and  0(1  +  /i)  =  ^^  _|_  2.  Accordingly, 
0(1  —  ^)  <  0(1)  <  0(1  +  /i),  and  thus  0(1)  is  not  a  turning  value  of  the 
function. 

Ex.  4.  Examine  the  functions  in  Exs.  4,  5,  Art.  75,  at  the  critical 
values  of  x. 

(6)  Examination  of  the  first  derivative  of  tlie  function.  When  a 
function  is  increasing,  its  derivative  is  positive  and  the  slope  of 
its  graph  is  positive ;  when  a  function  is  decreasing,  its  derivative 
is  negative  and  the  slope  of  its  graph  is  negative  (Art.  74).  Hence, 
h  being  taken  as  small  as  one  pleases,  if  <t>'(a  —  h)  is  positive  and 
(f>'{a  +  Ji)  is  negative,  then  <f>(a)  is  a  maximum  value  of  <f>(x).  On 
the  other  hand,  if  (f>'(a  —  h)  is  negative  and  <t>'{a  -f-  h)  is  positive, 
then  <^(.r)  is  decreasing  when  x  is  approaching  a,  and  <f>(x)  is 
increasing  when  x  is  leaving  a,  and  accordingly  ^(a)  is  a  mini- 
mum value  of  ^(x). 

Note  2.  Test  (b)  is  generally  easier  to  apply  than  test  (a).  For  test  (a) 
the  functions  0(a  —  h)  and  0(a  +  h)  must  be  computed  ;  for  test  (&)  merely 
the  algebraic  signs  of  0'(a  —  h)  and  0'(«t  +  h)  are  required, 

Ex.5,    (a)  InEx.l,  Art.  75. 0'f- —  ^  J  is  positive  and  0'f^  + /ij  is  nega- 
tive.    Accordingly,  0 (  -  ] ,  i-e.  sin  ^  or  1,  is  a  maximum  value  of  sin  x. 
(b)  Apply  this  test  at  the  other  critical  values  in  Ex.  1,  Art.  75. 

Ex.  6.  (a)  In  Ex.  2,  Art.  75^  0'(1  —  ^)  is  negative  and  0'(1  -}-  h)  is  posi- 
tive.    Accordingly  0(1),  i.e.  0,  is  a  minimum  value  of  (x  —  iy{x  -f  3). 

(b)  Apply  this  test  at  the  other  critical  value  in  Ex.  2,  Art.  75. 

Ex.  7.  In  Ex.  3,  Art.  75,  0'(1  —  h)  is  positive  and  0'(1  -|-  h)  is  positive. 
Accordingly,  0(1),  or  2,  is  neither  a  maximum  nor  a  minimum. 

Ex.  8.  Apply  test  (6)  at  the  critical  values  of  the  functions  in  Exs.  4,  5, 
Art.  75. 

(c)  Examination  of  tlie  second 'derivative  of  tlie  function.     It  has 

been  seen  that  the  sign  of  the  derivative  of  a  function  <^  (a*)  changes 
from 'positive  to  negative  when  the  function  is  passing  through  a 


122  INFINITESIMAL   CALCULUS.  [Ch.  VII. 

maximum  value.  If  the  derivative  <^'(.t)  passes  from  a  positive 
value  to  zero,  and  then  becomes  negative,  the  derivative  is  contin- 
ually decreasing,  and  hence  its  derivative,  namely  (f>"{x),  must  be 
negative  for  the  critical  value  of  x.  On  the  other  hand,  when  the 
function  passes  through  a  minimum  value,  the  derivative  changes 
sign  from  negative  to  positive.  If  then  the  derivative  cf>'(;x) 
passes  through  zero,  it  is  continually  increasing,  and  hence  its 
derivative,  namely  <^"(.^),  must  be  positive  for  the  critical  value 
of  X.  Therefore, 
if  <^'(a)  is  zero  and  <f>"(a)  is  negative,  (f>(a)  is  a  maximum  value 

of  ^{x); 
if  (ji'(a)  is  zero  and  cj)"(a)  is  positive,  <t>(a)  is  a  minimum  value 

of  (^(x). 

Note  3.  When  the  second  derivative  can  be  obtained  readily,  test  (c)  is 
the  easiest  of  the  three  tests  to  apply. 

Note  4.  Sometimes  (p"(a)  is  also  zero.  A  procedure  to  be  adopted  in 
this  case  is  discussed  in  Art,  181.  One  of  the  other  tests,  however,  may  be 
used. 

Note  5.  Historical.  Kepler  (1571-1630),  the  great  astronomer,  "was 
the  first  to  observe  that  the  increment  of  a  variable  —  the  ordinate  of  a  curve, 
for  example  — is  evanescent  for  values  infinitely  near  a  maximum  or  minimum 
value  of  the  variable."  Pierre  de  Fermat  (1601-1665),  a  celebrated  French 
mathematician,  in  1629  found  tlie  values  of  the  variable  that  make  an  expre.s- 
sion  a  maximum  or  a  minimum  by  a  method  which  was  practically  the 
calculus  method  (Art.  75). 

Note  6.  Many  problems  in  maxima  and  minima  may  be  solved  by  ele- 
mentary algebra  and  trigonometry.  For  the  algebraic  treatment  see  (among 
other  works)  Chrystal.  Ali/fhra,  Part  II.,  Chap.  XXIV.;  Williamson,  Diff. 
Cal.,  Arts.  133-137  ;    Gibson,  Calculus,  §  76  ;   Lamb,  Calculus,  Art.  52. 

Note  7.  Maxima  and  ininiina  of  functions  of  two  or  more  inde- 
pendent variables.  For  discussions  of  this  topic  see  McMahon  and  Snyder, 
Diff.  Cal.,  Chap.  X.,  pages  183-197;  Lamb,  Calculus,  pages  135,  596-598; 
Gibson,  Calculus,  §§  159,  160  ;  Echols,  Calculus,  Chap.  XXX.  ;  and  the 
treatises  of  Todhunter  and  Williamson. 

EXAMPLES. 

9.    (a)  In    Ex.    1,   Art.    75,    <p" (x)  =  -  sin  x.      Accordingly,    <P"l^j  is 
negative,  and  thus  0(  -  j?  i-^-  shi  -,  is  a  maximum  value  of  (P(x). 
(6)  Apply  test  (c)  at  the  other  critical  values  of  sinx. 


77.]  PROBLEMS    IN    MAXIMA    AND    MINIMA.  123 

10.  (a)  In  Ex,  2,  Art.  75,  0"(a;)=  2(3x^+ 1).  Accordingly,  0"(1)  is 
positive,  and  thus  0(1)  is  a  minimum  value  of  0(x). 

(b)  Apply  test  (c)  at  the  other  critical  value  in  Ex.  2,  Art.  75. 

11.  In  Ex.  3,  Art.  75,  0"(x)=  G(x  -  1).  Here  0"(l)=O,  and  thus 
test  ((')  fails  to  indicate  whether  0(1)  is  a  turnhig  value  of  <p{x).    (See  Note  4.) 

12.  Apply  test  (c)  at  the  critical  values  of  the  functions  in  Exs.  4,  5, 
Art.  75. 

Note  8.  Sketching  of  graphs.  The  ideas  discussed  in  Arts.  74-76  are  a 
great  aid  in  making  graplis  of  functions,  and  in  showing  what  is  termed  the 
march  of  a  function. 

13.  For  each  of  the  following  functions  find  the  critical  values  of  x, 
determine  the  maxinmm  and  minimum  values,  and  sketch  tlie  graphs : 
(1)  2  a:3  +  5  x2  -  4  a-  -f  2  ;  (2)  5  +  12  a:  -  x^  -  2  x^ ;  (3)  x\x  +  1)  (x  -  2)^  ; 
(4)  (:«-2)3(x  +  l)-2;   (5)  2  +  3(jc  -  4)1  + (x-4)l  ;  (0)  3  x5_125  x*5  +  2160  x  ; 

(7)  a^"  -  7  a;  +  6       g.    {x-Tf     (9)  a;ioga; ;  (10)  x' ;  (11)  2  sin2x  +  8  cos'^x  ; 
^  ^        X  -  10       '    ^  ^    (.X  +  2)2 '  ^  ^         o     ,  V     y 

(12)  sinxsin2ic;    (13)  x  cos  x. 

14.  Show  that  a  +  (x  —  c)"  is  a  minimum  when  x  =  c,  if  n  is  even  ; 
and  that  it  has  neither  a  maximum  nor  a  minimum  value,  if  n  is  odd. 

15.  (a)  Show  that  (ac  —  If-)  -^  a  is  a  maximum  or  a  minimum  value 
of  ax2  +  6x  +  c,  according  as  a  is  positive  or  negative.  (6)  Show  that 
ax^  +  6x  +  c  cannot  have  both  a  maximum  and  minimum  value  for  any 
values  of  a,  &,  c. 

16.  Find  the  point  of  maximum  on  the  curve  x'  +  y^  —  3  axy  =  0. 
Sketch  the  graph,  taking  a  =  1. 

17.  In  the  case  of  the  ellipse  ax-  +  2  hxy  +  hy'^  +  c  =  0,  show  how  to 
find  the  highest  and  lowest  points,  and  the  points  at  the  extreme  right  and 
left. 

77.  Practical  problems  in  maxima  and  minima.  Some  practical 
applications  of  the  principles  of  Arts.  75  and  76  will  now  be 
given.  In  making  these  applications  the  student  is  in  a  position 
analogous  to  his  position  in  algebra  when  he  applied  his  knowledge 
about  the  solution  of  equations  to  solving  "word  problems."  Here, 
as  in  algebra,  the  most  difficult  part  of  the  work  is  the  mathe- 
matical statement  of  the  problem  and  the  preparation  of  the  data 
for  the  application  of  the  processes  of  Art.  76. 


124 


INFINITESIMAL   CALCULUS, 


[Ch.  VII. 


EXAMPLES. 


1.  Find  the  area  of  the  largest  rectangle  that  can  be  inserted  in  a 
given  triangle,  when  a  side  of  the  rectangle  lies  on  a  side  of  the  triangle. 

Let  ABC  be  the  given  triangle,  and  let 
the  given  values  of  the  base  AB  and  the 
height  CD  be  b  and  h  respectively. 

Suppose  that  MQ  is  the  largest  rectangle, 

and  let  MN  and  NQ  be  denoted  by  y  and  x 

respectively,  and  denote  the  area  of  MQ  by  u. 

Then  u  =  xy,  which  is  to  be  a  maximum. 

It  is  first   necessary   to    express  m,   the 

quantity  to  be  "maximised,"  in  terms  of  a 

Fig,  32.  single  variable. 


M 

/ 

/ 

\ 

,P 

h 

A 

y 

1 

D 

i 

i          T        ) 

B 

u 

Now 


MP  :  AB  =  CH :  CD  :  i.e.  x:h 


h. 


x  =  -(h  —  y);  accordingly,  u  =  -  y(h  —  y),  a.  maximum. 
h  h 


(h-2 


0 ;    whence    y 


Thus    x  =  lb,    and    area 


.    du 

"  dy       h 
MQ  =  \bh  =  one  half  the  area  of  the  triangle. 

Note  1.  If  M  be  supposed  to  move  along  AC  from  A  to  C,  the  rectangle 
MQ  increases  from  zero  at  A  and  finally  decreases  to  zero  at  C.  It  is  thus 
evident  that  for  some  point  between  A  and  C  the  rectangle  has  a  maximum 
value. 

Note  2.  In  these  examples  it  is  necessary  that  the  quantity  to  be  maxi- 
mised or  minimised  be  expressed  in  terms  of  one  variable.  Conditions 
sufficient  for  this  must  be  provided. 

2.  Solve  Ex.  1,  expressing  u  in  terms  of  x. 

3.  A  parabola  y'^  =  8x  is  revolved  about  the  a;-axis ;  find  the  volume 
of  the  largest  cylinder  that  can  be  inscribed  in  the 

paraboloid  thus  generated,  the  height  of  the  parab- 
oloid being  4  units. 

Let  OPL  be  the  arc  that  revolves,  LN  be  at 
right  angles  to  OX,  and  ON  =  4.  Take  P(x,  y), 
a  point  in  OL,  and  construct  the  rectangle  PN. 
When  OPL  generates  the  paraboloid,  PN  gen- 
erates a  cylinder.  (As  P  moves  along  the  curve 
from  O  to  L,  the  cylinder  increases  from  zero  at 
O  and  finally  decreases  to  zero  at  L.  Thus  there 
is  evidently  some  position  of  P  between  0  and  L 
for  which  the  cylinder  is  a  maximum.)     Suppose  Fig.  33. 


Y 

5 

J 

^ 

r 

V 

N 

0 

X 

f7.] 


EXAMPLES. 


125 


Accordingly, 


ttFG'  .  GX 

8  7r(4-2a;)  =  0. 


that  PiV  generates  the  maximum  cylinder,  and  denote  its  volume  by   V. 
'^^^^^  V  =  irFG'  .  GX  =  7r?/2(4  -  x)  =  8  7rx(4  -  x). 

dx 
From  this,  x  =  2  ;  hence  F=  100.53  cubic  units. 

Note  3.  In  the  process  of  maximising  in  Exs.  1,  2,  the  constant  factors  - 
and  8  tt  may  as  well  be  dropped.     (See  Art.  76,  Note  1.) 

Note  4.  In  each  of  these  examples  it  is  well  to  perceive  at  the  outset  that 
a  maximum  or  a  minimum  exists. 

4.  A  man  in  a  boat  6  miles  from  shore  wishes 
to  reach  a  village  that  is  14  miles  distant  along 
the  shore  from  the  point  nearest  to  him.  He  can 
walk  4  miles  an  hour  and  row  3  miles  an  hour. 
AVhere  should  he  land  in  order  to  reach  the  village 
in  the  shortest  possible  time  ?  Calculate  this 
time.  Let  L  be  the  position  of  the  boat,  M  the 
village,  and  N  the  nearest  land  to  L.  Then  LN 
is  at  right  angles  to  NM.  Let  P  denote  the  place 
to  land,  and  T  denote  the  time  (in  hours)  to  go 
over  LP  +  PM^  and  denote  NP  by  x. 


NP^6.8 


Then 


LP     PM 
3  4 


V36  +  x^      U 


a  mmimum. 


dT 
dx 


3V36 


-^^  =  0. 
4 


Hence,  x  =6.8  miles,  and  r  =  4.8  •••  hours. 

5.    What  must  be  the  ratio  of  the  height  of  a  Norman  window  of  given 
perimeter  to  the  width  in  order  that  the  greatest  possible  amount  of  light  may 

be  admitted  ?     (A  Norman  window  consists 
of  a  rectangle  surmounted  by  a  semicircle.) 

Let  m  denote  the  given  perimeter,  2x  the 
width,  and  ij  the  height  of  the  rectangle  in  the 
window  desired  ;  let  A  denote  the  area  of 
the  window. 

Then  A  =  2xy  +  1  ttx^. 

Now  2x-^2y-\-'7rx  =  m. 

.'.  A  =  mx  —  2  x^  —  i  7rx2, 

which  is  to  be  a  maximum. 

On  finding  the  value  of  x  for  which  ^  is  a 
maximum,  and  then  getting  the  corresponding  value  of  y,  it  will  appear  that 
x  =  y.    Accordingly,  the  height  MD  =  the  width  AB. 


E 

1 

C 

I 

M 

4 

B 

Fig.  35. 


126  INFINITESIMAL    CALCULUS.  [Ch.  VII. 

6.  Find  the  area  of  the  largest  rectangle  that  can  be  inscribed  in  an 
ellipse.     (First  show  that  there  evidently  is  such  a  rectangle.) 

Suggestions  :  Let  the  semiaxes  of  the  ellipse  be  a  and  6,  and  choose 
axes  of  coordinates  coincident  with  the  axes  of  the  ellipse.    Let  P(ic,  y)  be  a 

vertex  of  the  rectangle.     Then  area  rectangle  =ixy  =  i  -xVa:^  —  x-.     Maxi- 

a 
mise  the  last  expression,  or,  better  still,  because  it  is  easier  to  do,  maximise 
the  square  of  xVa^  -  x;\  viz.  a;2(a"2  -  x^).     (See  Art.  76,  Note  1.)     It  will  be 
found  that  the  area  of  the  rectangle  is  2  aft,  half  the  area  of  the  rectangle 
circumscribing  the  ellipse. 

7.  Divide  a  number  into  two  factors  such  that  the  sum  of  their  squares 
shall  be  as  small  as  possible. 

8.  Two  sides  of  a  triangle  are  given :  find,  by  the  calculus,  the  angle 
between  them  such  that  the  area  shall  be  as  great  as  possible. 

9.  Find  the  largest  rectangle  that  can  be  inscribed  in  a  given  circle. 

10.  Through  a  given  point  P(a,  6)  a  line  is  drawn  meeting  the  axes 
in  A  and  J5 ;  0  is  the  origin  :  Find  (i)  the  least  length  that  AB  can  have  ; 
(ii)  the  least  value  of  OA  +  OB ;  (iii)  the  least  possible  area  of  the  triangle 
OAB. 

11.  A  and  B  are  points  on  the  same  side  of  a  straight  line  MN : 
determine  the  position  of  a  point  C  in  MN'.  (1)  so  that  AG^  +  CB"  =  a 
minimum  ;  (2)  so  that  AC  +  CB  =  a.  minimum. 

N,B.    The  cones  and  cylinders  in  the  following  examples  are  right  circular  : 

12.  (i)  Find  the  height  of  the  cone  of  greatest  volume  that  can  be  in- 
scribed in  a  sphere  of  radius  r.  (ii)  Find  the  cone  of  greatest  convex  surface 
that  can  be  inscribed  in  this  sphere. 

13.  Find  the  semi-vertical  angle  of  the  cone  of  least  volume  that  can  be 
described  about  a  sphere. 

14.  (i)  Find  the  cylinder  of  greatest  volume  that  can  be  inscribed  in  a 
sphere  of  radius  r.  (ii)  Find  the  cylinder  of  greatest  curved  surface  that 
can  be  inscribed  in  this  sphere. 

15.  (i)  Determine  the  maximum  cylinder  that  can  be  inscribed  in  a 
right  circular  cone  of  height  h  and  radius  of  base  a.  (ii)  Determine  the 
cylinder  of  greatest  convex  surface  that  can  be  inscribed  in  this  cone. 

16.  What  is  the  ratio  of  the  height  to  the  radius  of  an  open  cylindrical 
can  of  given  volume,  when  its  surface  is  a  minimum  ? 

17.  A  circular  sector  of  given  perimeter  has  the  greatest  area  possible: 
find  the  angle  of  the  sector. 

18.  It  is  required  to  construct  from  two  circular  iron  plates  of  radius 
a  a  buoy,  composed  of  two  equal  cones  having  a  common  base,  which  shall 
have  the  greatest  possible  volume  :  find  the  radius  of  the  base. 


78.] 


POINTS    OF   INFLEXION. 


127 


19.  An  open  tank  of  assigned  volume  has  a  square  base  and  vertical 
sides  :  if  the  inner  surface  is  the  least  possible,  what  is  the  ratio  of  the  depth 
to  the  width  ? 

20.  From  a  given  circular  sheet  of  metal  it  is  required  to  cut  out  a 
sector  so  that  the  remainder  can  be  formed  into  a  conical  vessel  of  maximum 
capacity  :  show  that  the  angle  of  the  sector  removed  must  be  about  66°. 

21.  In  a  submarine  telegraph  cable  the  speed  of  signalling  varies  as 
x^  log  -,  where  x  is  the  ratio  of  the  radius  of  the  core  to  that  of  the  covering : 

show  that  the  speed  is  greatest  when  the  radius  of  the  covering  is  Ve  times 
the  radius  of  the  core. 

22.  Assuming  that  the  power  required  to  propel  a  steamer  through  still 
water  varies  as  the  cube  of  the  speed,  find  the  most  economical  rate  of 
steaming  against  a  current  which  is  running  at  a  given  rate. 

23.  Assuming  that  the  strength  of  a  rectangular  beam  varies  as  the 
product  of  the  breadth  and  the  square  of  the  depth  of  its  cross-section,  find 
the  breadth  and  depth  of  the  strongest  rectangular  beam  that  can  be  cut  from 
a  cylindrical  log,  the  diameter  of  whose  cross-section  is  d  inches. 

24.  Find  the  length  of  the  shortest  beam  that  can  be  used  to  brace  a 
vertical  wall,  if  the  beam  must  pass  over  another  wall  that  is  a  feet  high  and 
distant  b  feet  from  the  first  wall. 

25.  At  what  distance  above  the  centre  of  a  circle  of  radius  a  must  an 
electric  light  be  placed  in  order  that  the  brightness  at  the  circumference  of 
the  circle  may  be  the  greatest  possible  ?  (Assume  that  the  brightness  of  a 
small  surface  A  varies  inversely  as  the  square  of  the  distance  r  from  a  source 
of  light,  and  directly  as  the  cosine  of  the  angle  between  ?•  and  the  normal  to 
the  surface  at  A.)     (Gibson's  Calculus.) 

78.  Points  of  inflexion:  rectangular  coordinates.  As  a  point 
moves  along  the  curve  LAM  from  L  to  3f,  the  tangent  at  the 
moving  point  changes  from  the  position  shown  at  L  to  that  at  A 

r 


o 

Fig.  36  b. 


and  then  to  that  at  M.  In  going  from  the  position  at  L  to  the 
position  at  A,  the  tangent  turns  in  the  direction  opposite  to  that 
in  which  the  hands  of  a  watch  revolve ;  in  going  from  the  position 


128  INFINITESIMAL   CALCULUS.  [Ch.  VII. 

at  A  to  the  position  of  M,  the  tangent  turns  in  the  same  direction 
as  that  in  which  the  hands  of  a  watch  revolve.  Points  such  as 
A,  D,  H,  G  (Fig.  36),  and  Q,  P,  R,  S  (Figs.  31  a,  b),  at  which  the 
tangent  for  the  point  moving  along  the  curve  ceases  to  turn  in 
one  direction  and  begins  to  turn  in  the  opposite  direction,  are 
called  points  of  inflexion. 

Examination  of  the  curve  for  points  of  inflexion.     As  the  moving 

point  goes  along  the  curve  from  L  to  A,  -^  increases  and  accord- 

dhi  ^^ 

ingly  its   derivative  -^,  is  positive ;  as  the  moving  point  goes 

along  the  curve  from  ^  to  If,  ^  decreases,  and  accordingly  J 

dx  ,2  (^'^ 

is  negative.    Thus  in  the  case  of  the  curve  LAM,  —-  is  positive  on 

one  side  of  A  and  negative  on  the  other.    Now  -^  changes  continu- 

d-v       ^^ 
ously  from  L  to  M-,  accordingly,  at  A  —^~  =  0.      Hence,  in  order 

dxr 

to  find  the  points  of  inflection  for  a  curve  y  =  f{x),  proceed  as 
follows : 

Calculate  T"^  j 

d^v 
then  solve  the  equation  — ^  =  0. 

axr 

This  will  give  critical  values  (or  points)  which  are  to  be  further 
examined  or  tested.     A  critical  point  is  tested  by  finding  whether 

— ^  has  opposite  signs  on  each  side  of  the  point.  If  — ^  has  oppo- 
dx-  p  dx^ 

site  signs,  the  critical  point  is  a  point  of  inflexion;  if  ~  has  the  same 

ax" 

^^_       ^^^    sign   on   both   sides  of  the   critical 

^^       ""^^  point,  as  in  Fig.  36  c,  the  point  is 

^^^'      ^'  what  is  called  a  point  of  undulation. 

Note  1.  At  a  point  of  inflexion  the  tangent  crosses  the  curve.  The  tan- 
gent at  an  ordinary  point  on  a  curve  is  the  limiting  position  of  a  secant  when 
two  of  the  points  of  intersection  of  the 
secant  and  the  curve  become  coincident 
(Art.  24).  The  tangent  at  a  point  of  in- 
flexion is  the  limiting  position  of  a  secant 
which  cuts  the  curve  in  more  than  two 
points,  when  the  secant  revolves  until  three 
points   of   intersection  become  coincident. 


78.]  EXAMPLES.  ^        129 

Thus  FT,  the  tangent  at  the  point  of  inflexion  J*,  is  the  limiting  position 
of  the  secant  MPQ  when  MPQ  revolves  about  P  until  31  and  Q  simultane- 
ously coincide  with  P.  At  a  point  of  undulation  the  tangent  does  not  cross 
the  curve.  The  tangent  at  a  point  of  inflexion  is  called  an  inflectional  tan- 
gent ;  the  tangent  where  y"  =  0  is  called  a  stationanj  tangent. 

Note  2.  If  f(x)  is  a  rational  integral  function  of  degree  w,  the  greatest 
number  of  points  of  inflexion  that  the  curve  y=f(x)  can  have  is  n  —  2. 
Moreover  the  points  of  inflexion  occur  between  points  of  maxima  and  minima. 
[See  F.  G.  Taylor's  Calculus  (Longmans,  Green  &  Co.),  Art.  206.] 

Note  3.  References  for  collateral  reading.  On  maxima  and  minima  of 
functions  of  one  variable,  etc. :  McMahon  and  Snyder,  Diff.  Cal.,  Chap.  VI. ; 
Echols,  Calculus,  Chap.  VIII.  (in  particular,  Art.  85).  On  points  of  inflexion : 
Williamson,  Diff.  Cal.  (7th  ed.).  Arts.  221-224  ;  Edwards,  Treatise  on  Diff. 
Cal.,  Arts.  274-279;  Echols,  Calculus,  Chap.  XI. 

Note  4.  Points  of  inflexion :  polar  coordinates.  For  a  discussion  of 
this  topic  see  Todhunter,  Dif.  Cal.,  Art.  294;  Williamson,  Diff.  Cal., 
Art.  242;   F.  G.  Taylor,   Calculus,  Art.  276. 

EXAMPLES. 

1.  In  the  following  curves  find  the  points  of  inflexion,  and  write  the 
equations  of  the  inflexional  tangents  ;  also  sketch  thv3  curves  and  draw  the 
inflexional  tangents  :    (I)  y  =  x^ ;  (2)  x  -  S  =  (y  +  Sy  ;  (3)  y  =  x-(i  -  x)  ; 

(4)   122/  =  x3-6x2  +  48;    (5)  y^-~-.\    (6)  y=~^\    (7)  V  ^  j^^' 

2.  Find  the  points  of  inflexion  on  the  following  curves :  (1)  y  = 
x(x  -  ay  ;     (2)    xy'^  =  a^(a  -  x)  ;     (3)    ax^  -  x^y  -a^y  =  0;     (4)   y  =  6  + 

(c-x)3;    (5)   y  =  m-b(x-cy;    (6)   x^  -  d  bx^ -\- a^y  =  0. 

3.  Show  that  the  curve  y  =  x^  has  no  point  of  inflexion.  Sketch  the 
curve. 

4.  Show  that  the  points  where  the  curve  y  =  bsin  -  meets  the  x-axis 
are  all  points  of  inflexion.  ^ 

5.  Show  that  the  curve  (1  +  xYy  =  1  —x  has  three  points  of  inflexion, 
and  that  they  lie  in  a  straight  line. 

6.  Show  why  a  conic  section  cannot  have  a  point  of  inflexion. 

7.  Show,  both  geometrically  and  analytically,  why  points  of  inflexion 
may  be  called  points  of  maximum  or  minimum  slope. 


CHAPTER   VIII. 

DIFFERENTIATION  OF   FUNCTIONS   OF   SEVERAL 
VARIABLES. 


N.B,  This  chapter  may  be  studied  immediately  after  Chapter  VII.,  or 
its  study  may  be  postponed  and  taken  up  after  any  one  of  Chapters  IX.-XX.* 

79.  Partial  derivatives.  Notation.  Thus  far  functions  of  one 
independent  variable  have  been  treated;  functions  of  two  and 
of  more  than  two  independent  variables  will  now  be  considered. 

I^et  u=f{x,y)  (1) 

in  which  f(x,  y)  is  a  continuous  function  (see  Note  2)  of  two 
independent  variables  x  and  y.  The  value  of  the  function  for  a 
pair  of  values  of  x  and  y  is  obtained  by  substituting  these  values 
in  fix,  y). 


Thus,  if  /(x,  2/)  =  3  X 
Z 


2  2/  +  7,  /(I,  2)  :=  3  .  1  -  2  .  2  +  7  =  6. 


Fig.  38. 


Note  1.  Geometrical 
representation  of  a  func- 
tion   of    two    variables. 

The  student  knows  how  a 
continuous  function  of  one 
variable  can  be  represented 
by  a  curve.  A  continuous 
function  of  two  variables 
can  be  represented  by  a  sur- 
face. Thus  the  function  z^ 
^hen     ,=/(^,j,),  (2) 

is  represented  by  the  sur- 
face LEGS  if  MP,  the  per- 
pendicular to  the  a;y-plane 
erected  at  any  point  M{x,  y) 
on  that  plane  and  drawn  to 
meet  the  surface  at  P,  is 
equal  to /(a;,  y). 


*  See  the  order  of  the  topics  in  Echols'  Calculus. 
130 


79.]  PABTIAL    DERIVATIVES.  131 

References  for  collateral  reading.  See  chapters  on  the  geometry  of 
three  dimensions  in  text-books  on  Analytic  Geometry,  for  instance,  those  of 
Tanner  and  Allen,  Ash  ton,  Wentworth  ;  also  Echols'  Calculus^  Chap.  XXIV. 

Note  2.  Continuous  function  of  two  variables  defined.  A  function 
/(x,  y)  is  said  to  be  a  continuous  function  of  x  and  y  within  a  certain  range 
of  values  of  x  and  y,  when  :  (i)  /(x,  y)  does  not  become  infinitely  great,  and 
(ii)  if,  (a,  b)  and  (a  -\-  h,  b  -\-  k)  being  any  values  of  (x,  y)  within  this 
range,  /(a  -\-  h,  b  -{-  k)  can  be  made  to  approach  as  nearly  as  one  pleases  to 
/(«,  6)  by  diminishing  h  and  k,  and  if  /(a  -\-h,b  +  k)  becomes  equal  to  /(a,  6), 
no  matter  in  what  way  h  and  k  approach  to,  and  become  equal  to,  zero. 
This  definition  may  be  illustrated  geometrically,  thus  :  On  the  X!/-plane 
(Fig.  38)  let  M  be  (a,  b)  and  N  he  (a  +  h,  b  +  k),  and  let  i¥P  be  /(a,  b) 
and  NQ  be  /(a  -^  h,  b  +  k).  Then,  if  MP  and  NQ  are  finite,  and  if  NQ 
remains  finite  while  N  approaches  31,  and  becomes  equal  to  MP  when  iV 
reaches  M,  no  matter  by  what  path  of  approach  on  the  xy-plane,  /(x,  y)  is 
said  to  be  a  continuous  function  of  x  and  y  iov  x  =  a  and  y  =  b. 

In  (1)  suppose  that  x  receives  a  change  Aa?  and  that  y  remains 
unchanged.     Then  u  receives  a  corresponding  change  Aw,  and 

u -\- Au  =  f(x  +  Ax,  y)', 

and  Au  =  f(x  +  Ax,  y)  -  f(x,  y). 

.  Aw  ^  f(x-^  Ax,  ?/)  -f(x,  y)^ 
' '  Ax  Ax 

,  T  Aw      T  fix  4-  Ax,  y)  —  f(x,  y) 

Ax  Ax 

This  limiting  value  is  called   the  partial   derivative   of  u  with 
respect  to  x,  because  there  is  a  like  derivative  of  u  with  respect 

to  y,  namely,     lim.,.„  ^  =  lim.,.o  /^  .^  +  Ay)  - /(a.,  y) . 
"Ay  Ay 

These  partial  derivatives  are  usually  written 

^,     ^,  (3) 

doc      dy 

respectively,  in  order  to  distinguish  them  from  derivatives  (like 

dM^  du^  ds^  ^^^  g^      s  ^^  functions  of  a  single  variable  and  from 

dx    dy    dt 

what  are  called  total  derivatives  (see  Art.  81).     If  u  =f(x,  y,  z), 


132  INFINITESIMAL   CALCULUS.  [Ch.VIII. 

the   partial   derivatives  of  the  first  order  are   — ,  — ,  and  — • 

dx    dy  dz 

According  to  the  above  definition,  the  partial  derivative  with 
respect  to  each  variable  is  obtained  by  differentiating  the  func- 
tion as  if  the  other  variable  were  constant.  Notation  (3)  is  very 
commonly  used,  but  various  other  symbols  for  partial  derivatives 
are  also  employed. 

Note  3.    Geometrical  representation  of  partial  derivatives  of  a  func- 
tion of  two  variables.     Let  f(x,  tj)  be  represented  by  the  surface  LEGS 

(Fig.  38)  whose  equation  is  _  ..     ^ 

^  —J  K^^  y)' 

Take  P  any  point  (x,  ?/,  z)  on  this  surface.  Through  P  pass  planes  parallel 
to  the  planes  ZOXand  ZOY,  and  let  them  intersect  the  surface  in  the  curves 
LPG  and  PP*S'  respectively.  Along  BPS^  x  remains  constant ;  and  along 
LPG,  y  remains  constant.      Accordingly,  from   the  definition  above  and 

Art.    24   the   partial   x-derivative    —  is  the  slope  of  LPG  at  P,  and  the 


dz 


dx 


partial  ^/-derivative  i^  is  the  slope  of  EPS  at  P. 
dy 

EXAMPLES. 

1.  If  u  =  x^  +  2  x^y  +  xy^  -{-  y^  -\-  e^  +  x  cos  ?/, 
then                       ^  =  3  a;2  -f  4  x?/  4-  ?/^  +  e^  +  cos  y, 

and  ^  =  2x'^-{-Sxy^  +  4:i/-xsmy. 

dy 

2.  Find  ^,  ^,  and  ^,  when  u=x^^2y^+S z--{-e'' sin  y+cos z cosy. 

dx  dy  dz 

3.  On  the  ellipsoid  ^-^Vl  +^  =  \:    (a)  find  ^  and  ^  at  the  point 

^  16     25      9  ^  ^  dx  dy 

where  x=l  and  y  =  4: ;    (b)  find  —  and  —  at  the  point  where  y  =  2  and 

z  =  2  :    (c)  find  ^  and  ^  at  the  point  where  z  =  I  and  x  =  S.     Make 

'    ^  ^  dz  dx 

figures  for  (a),  (ft),  and  (c),  and  show  what  these  partial  derivatives  repre- 
sent on  the  ellipsoid. 

4.  Verify  the  following  : 

(i)  ltu  =  \og(ie--^ey),  |l*  +  |^  =  l; 
dx     dy 

^   ^  e-4-ey'  dx     dy 

(iii)  If  M  =  xyy,  x^  +  y^  =  (x-\-y  +  \og u)ii. 
dx        dy 


80.]  SUCCESSIVE    PARTIAL    DERIVATIVES.  133 

80.  Successive  partial  derivatives.  The  partial  derivatives  of 
the  first  order  described  in  Art.  79  are,  in  general,  also  continuous 
functions  of  the  variables,  and  their  partial  derivatives  may  also 
be  required.  In  the  successive  differentiation  of  functions  of  two 
or  more  variables,  the  following  is  one  of  the  systems  of  notation : 


—  I—]      IS  written  — - : 
dx\dxj                         dx'' 

aUS'-"""?^' 

±(^^]      is  written    ^'^   ; 
dy\dxj                         dydx' 

dx\dyj                      dxdy 

dz\dydxj                      dzdydx 

d  fdhC\  .         .^^        dhi 

■  dz\dxdzj                     dzdxdz' 

dz\df)                    dzdf 

and  so  on. 

Note  1.  In  this  notation  the  symbol  above  the  horizontal  bar  indicates 
the  order  of  the  derivative,  and  the  symbols  below  the  bar,  taken  from  right 
to  left,  indicate  the  order  in  which  the  successive  differentiations  are  to  be 

performed.     Thus  — ^-^ —  means  that  u  is  to  be  differentiated  three  times 

dx^  dy  dz^ 
in  succession  w^ith  respect  to  z^  and  the  result  is  then  to  be  differentiated 
with  respect  to  ?/ ;  and  the  function  thus  obtained  is  then  to  be  differentiated 
twice  in  succession  with  respect  to  x. 

Note  2.  The  adoption,  by  mathematicians,  of  the  symbol  d  in  the  nota- 
tion of  partial  differentiation  was  mainly  due  to  the  great  mathematician, 
Carl  Gustav  Jacob  Jacobi  (1804-1851),  who  decided,  in  1841,  to  use  d  in 
the  manner  which  afterwards  became  the  fashion.  As  to  some  points  of 
insufficiency  and  difficulty  connected  with  this  notation,  see  correspondence 
between  Thomas  Muir  and  John  Perry,  Nature,  Vol.  66,  pages  53,  271,  520. 

Note  3.  The  order  in  which  the  snccessive  differentiations  are  per- 
formed does  not  affect  the  result  {certain  conditions  being  satisfied)  ;  e.g. 

d'^n  _  d'U  d^n  d^n      _     d^i  d^u     _    d^u    _    d^u 


dxdy     dydx    dxdydz     dzdxdy     dydxdz    dzdxdz     dz^dx     dxdz^ 

This  theorem  is  discussed  in  Art.  85  ;  it  may  be  verified  in  the  examples 
that  follow,  especially  in  the  important  example,  Ex.  1.  For  a  simple 
example  illustrating  an  unwarrnnted  assumption  sometimes  made  in  the 
proof  of  this  theorem,  see  Gibson,  Calculus^  page  221. 


134  INFINITJESIMAL   CALCULUS.  [Ch.VIII. 


EXAMPLES. 

1.  Show  that  —^ — {Ax"^y'^)  =  —^ —  (^x"*//"),   in  which  A,   m,   and    n 

dxdy  dydx  ^<^  ^. 

are  constants.     Then  show  that  if  w  =  S J.x'^^'*,    — —  =  — ,  and  hence 

5x  by     dy  dx 
that  the  theorem  in  Note  3  is  true  for  all  algebraic  functions. 

2.  In    the    following    instances    verify    the    fact    that    -^ —  =  — — ; 

V  av-hx  ^^^^      ^^^^ 

u  =  sin  (xy)  ;  u  =  cos^;  u  =  xm ;  u  =  -^ —  ;  u  =  sec  {ax  -^  by)  ;  u  =  xlogy] 

X  by  —  ax 

u  =  xsiny  -\-  ysinx;  u  =  y  log  (1  +  xy)  ;  m  =  sin  (x^)  ;  u  =  sin  (a:)^. 


3.    In   the    following    instances  verify   that 


B^u    _      d^'it      _    d^u 


/,,x  ^  ,  ,,     v/5x     dydxdy     dxdy^ 

(i)  u  =  a  tan-i  M^    ;  (ii)  u  =  sin  (xy)  +  :^^t_^. 
\x/  xy 

4.    Show  that     ^^^^     =     ^  ^    ,  when  u  =  cos  (ax"-  +  by""). 


dx^  dy'^     dy'^  dx^ 


I    .u^„,  +!,„.  d-u  _  ^2d^. 


5.  If  u  =  tan  (y  +  ax) -\-{y  —  ax)^,  show  that  - —  =  a 

6.  If   h:-^^,   show  that  x^  +  y-^^  =  2^,   and    that   2/f^^  + 

x  +  y  dx^         dx  By        dx  dy^ 

B'u   _  2^. 

5x  By       By 


7.  If  w  =  Vx2  +  ?/2,  show  that  x2  ^LL^  +  2  xy  -^  +  ^2  y_L^  ^  _  ^  w. 

5X2  5a;  ^^  ^1,2  9 

8.  If  w  =  (x2  +  ^/2  +  ^2)-^,  show  that  ^  +  ^'  +  ^'  =  0. 

dx^      By^     Bz^ 

9.  Show  that  a  function  of  two  independent  variables  has  w  +  1  partial 
derivatives  of  order  n. 

81.   Total  rate  of  variation  of  a  function  of  two  or  more  variables. 

N.B,     Before  reading  this  article  and  the  next  it  is  advisable  to  review 
Arts.  25,  26. 

Given  that  u  =f(x,  y),  (1) 

and  that  x  and  y  vary  independently  of  each  other,  it  is  required 
to  find  the  rate  of  variation  of  u  in  terms  of  the  rates  of  variation 

of  X  and  y ;  i.e.  to  find  —  in  terms  of  -—  and  -^• 
"^^  dt  dt  dt 

In  (1)  let  X  and  y  receive  increments  Ax  and  Ay  respectively,  in 
a  time  At  say  ;  then  tt  receives  a  corresponding  increment  Au,  and 

u  -\-  Au  =f(x  +  Ax,  y  4-  Ay). 

.:  Au  =f(x  -\-Ax,y  +  Ay)  -f(x,  y).  (2) 


81.]  TOTAL    RATE    OF    VARIATION.  135 

Hence,  on  introduction  of  —f(x,y-\-Ay)-\-f(x,y-\-^y)  and 
division  by  A^, 

An  ^  f(x  4-  Ax,  y  +  Ay)  -f{x,  y  +  Ay)      f(x,  y  +  Ay)  -f(x,  y) 
At  At  At  . 

^  f(^ + Ax,  y  4-  Ay)  -f(x,  y + ^y)  .  Aa;  ^  f(x,y-\-Ay)-f(x,y)  ^  Ay^ 
Ax  At  Ay  At 

Now  let  At  approach  zero ;  then  Ax  and  Ay  approach  zero,  and, 
moreover  (if  a  certain  condition  is  satisfied), 

lijn  /(a;  +  Aa;,  y  +  Ay)  -/Q^',  y  +  Ay)  _  df(x,  y)  *         du^ 

and  lim^,^  /(x,  y+Ay)-/(x,  y)  ^  du^ 

Ay  By 

Hence,  ^  ^  ^w  cZ^     ^  f?y .  ^3^ 

'  dt      dx  dt      du  dt  ^  ^ 

In  words :  The  total  rate  of  variation  of  a  function  ofx  and  y  is 
equal  to  the  partial  x-derivative  multiplied  by  the  rate  of  variation  of 
X  plus  the  partial  y-derivative  multiplied  by  the  rate  of  variation  of  y. 

Similarly,  if  u  =f(x,  y,  z), 

du _du  dx     du  dy     du  dz  ,.. 

'di~'dx~dt      'dydi      dzdi'  ^  ^ 

Kesults  (3)  and  (4)  can  be  extended  to  functions  of  any  num- 
ber of  variables.  (All  derivatives  herein  are  assumed  to  be  con- 
tinuous.) 

Note  1.  A  function  may  remain  constant  while  its  variables  change. 
The  total  rate  of  variation  of  such  a  function  is  evidently  zero.    (See  Art.  84.) 

Note  2.  Suppose  that  in  (1)  y  is  a  function  of  x  and  that  the  derivative 
of  u  with  respect  to  x  is  required.  This  may  be  obtained  either  directly,  as 
(3)  has  been  obtained,  or  by  substituting  x  for  t  in  (3) ;  then 

du_du,du^,  (K\ 

due     doc     dy  dx  ^   ' 

Result  (5)  may  also  be  obtained  by  dividing  both  members  of  (3)  by  — 
[Art.  34(3)].  ^^ 

*  For  a  discussion  of  the  condition  necessary  and  sufficient  for  the  passage 
of  the  first  member  of  this  equation  into  the  second,  see  W.  B.  Smith,  Infini- 
tesimal Analysis,  Vol.  I,  Art.  205  (and  also  Arts.  206,  207). 


136  INFINITESIMAL   CALCULUS.  [Ch.VIII. 

Qu 
Note  3.    In  (5)  -^  is  the  x-derivative  of  u  when  y  is  treated  as  a  con- 

stant,  and  —  is  the  ic-derivative  of  u  when  y  is  treated  as  a  function  of  x. 
dx 

Here  —  is  called  the  total  ic-derivative  of  u, 

dx 

Similarly  the  total  ^/-derivative  ^^du^dudx^ 

dy      dy      dx  dy 


EXAMPLES. 

1.  Express  result  (5)  in  words. 

2.  Given  z  =  Sx'^-\-4y%  (1) 

fl^  (1  "Y  fill 

find  —  whenx=3,  ?/=— 4,  —  =  2  units  per  second,  and-^  =  3  units  per  second. 
dt  dt  dt 

On  d iff erentiation  in  ( 1 ),  ^  =  6  x  —  +  8  ?/  ^^  =  -  60. 
dt  dt  dt 

Geometrically  this  means  that  on  the  surface  (1),  which  is  an  elliptic 
paraboloid,  if  a  point  moves  through  the  point  (3,  —4,  91)  in  such  a  way 
that  the  x  and  y  coordinates  of  the  moving  point  are  there  increasing  at  the 
rates  of  2  and  3  units  per  second  respectively,  then  the  ^-coordinate  of  the 
moving  point  is,  at  the  same  place  and  moment,  decreasing  at  the  rate  of 
60  units  per  second. 

N.B.     Figures  should  be  drawn  for  Ex.  2  and  the  following  examples. 

3.  In  Ex.  3  (a),  Art.  79,  find  how  the  ^-coordinate  i«  changing  when 
the  ic-coordinate  is  increasing  at  the  rate  of  1  unit  per  second,  and  the 
^/-coordinate  is  decreasing  at  the  rate  of  2  units  per  second. 

4.  In  Ex.  3  (6),  Art.  79,  find  how  x  is  behaving  when  y  is  decreasing 
at  the  rate  of  2  units  per  second,  and  z  is  increasing  at  the  rate  of  3  units 
per  second. 

82.  Total  differential.  Let  dx  and  dy  be  differentials  of  the  x 
and  y  in  (1)  Art.  81.    They  may  be  regarded  as  quantities  such  that 

,       ,       dx   dy 
dx:  dy  =  — : -^ • 
dt    dt 

Now  let  du  be  taken  so  that 

du  =  ^dx  +  ^dy.  (1) 

doc  dy  ^  ^ 

As  used  in  (1)  -^  dx  is  called  the  partial  x-differential  ofu^  —dy 
dx  dy 

is  called  the  partial  y-differential  of  u,  and  du  is  called  the  total 
differential  of  u,  and  the  complete  differential  of  u. 


82.]  TOTAL    DIFFERENTIAL.  137 

Note  1.  When  y  is  a  function  of  x,  relation  (1)  follows  directly  from 
Eq.  (5),  Art.  81,  and  definition  (5),  Art.  27. 

Note  2.  The  partial  differentials  in  (1)  are  also  denoted  by  d^u  and  dyii, 
and  thus  (1)  may  be  written    a,,  ^^.^  ^  ^yU. 

Note  3.  In  general  the  du  in  (1)  is  not  exactly  equal  to  the  actual  change 
in  u  due  to  the  changes  dx  and  dy  in  x  and  y  ;  but  the  smaller  dx  and  dy  are 
taken,  the  more  nearly  is  du  equal  to  the  real  change  in  u  (see  exercises  below). 
The  differential  du  may  be  regarded  as,  and  is  very  useful  as,  an  approxima- 
tion to  the  actual  change  in  u.  In  some  cases  this  change  can  be  calculated 
directly  ;  in  others  it  can  be  found  to  as  close  an  approximation  as  one  pleases 
by  a  series  developed  by  means  of  the  calculus.  [See  Chap.  XX.,  in  par- 
ticular, Art.  176,  Eq.  (10),  and  Art.  178,  Note  5.] 

EXAMPLES. 

1.  Express  relation  (1)  in  words. 

2.  Given  it  =  3  a;2  -f  2  y'^,  find  du  when  x  =  2,  y  =  S,  dx  =  .01,  and 
dy  =  .02. 

Here  du  =  Qxdx -{- 4ydy  =  .12 -\-  .24:  =  .36. 

The  actual  change  in  m  is  3(2  •  01)2  _,_  2(3  .  02)2  _  (3  .  22  +  2  •  32)  =  .3611. 

3.  As  in  Ex.  2  when  dx  =  .001  and  dy  =  .002.    Also  find  the  change  in  u. 

4.  Find  the  complete  differential  of  each  of  the  following  functions  : 

(i)  tan-il^;  (ii)  y' ;  (iii)  x^f ;  (iv)  loga:y;  (v)  M  =  xiogy. 

5.  Find  dy  when  y  =  8  cos  A  sin  B,  A  =  40°,  dA  =  30',  B  =  65°, 
dB  =  20'. 

Note  4.  It  may  be  said  here  that  if  LEGS  (Fig.  38)  be  the  surface 
z  =/(x,  y),  and  if  M  be  {x,  y)  and  iV  be  (x  +  dx,  y  +  dy),  and  NQ  be  pro- 
duced to  meet  in  ^1  the  plane  tangent  to  the  surface  at  P,  then  the  total 
differential  dz  is  equal  to  JVQi  —  MP. 

Ex.     Prove  this  statement.     (Suggestion  :  make  a  good  figure.) 

Similarly  to  (1),  if  u  =f(x,  y,  z),  and  dx,  cJy,  dz,  be  differentials 
of  X,  y,  z,  respectively,  and  if  du  be  taken  so  that 

doc  dy  dz  ^  >' 

du  is  called  the  total  differential  of  u.     Relation  (2)  is  also  written 

du  =  d^u  -\-  dyU  -f  dgU. 

Definitions  (1)  and  (2)  may  be  extended  to  functions  of  any 
number  of  variables. 


138  INFINITESIMAL    CALCULUS.  [Ch.VIII. 

6.  Given  u  =  x^  -\-  y"^  -\-  2  z,  find  du  when  x  =  2,  ?/  =  3,  5;  =  4,  dx  =  .1, 
dy  —  .4,  dz  ——  .2).     Also  find  the  actual  change  in  u. 

7.  The  numbers  r«,  x,  ?/,  and  2;  being  as  in  Ex.  6,  dx  —  .01,  dy  =  .04,  and 
dz  =  —  .03,  calculate  the  difference  between  du  and  the  actual  change  in  u. 

8.  Find  du  when  w  =  x^". 

83.  Approximate  value  of  small  errors.  A  practical  application 
of  relations  (1)  and  (2)^  Art.  82,  may  be  made  to  the  calculation 
of  approximate  values  of  small  errors.  The  ideas  set  forth  in  the 
first  part  of  Art.  65  may  be  applied  to  any  number  of  variables. 

If  u  =  f(x,ij,z,---), 

and  dx,  dy,  dz,  •••,  be  regarded  as  errors  in  the  assigned  or  measured 
values  of  x,  y,  z,  •••,  then 

du  = —-  dx -\- —-  dy  -\ — -  dz-\ 

ox  dy    '       dz 

is,  approximately,  the  value  of  the  consequent  error  in  the  com- 
puted value  of  u.  Illustrations  can  be  obtained  by  adapting 
Exs.  2,  3,  5,  6,  7,  Art.  82.  In  applying  the  calculus  to  the  com- 
putation of  approximate  values  of  errors  it  is  usual  to  denote  the 
errors  (or  differences)  in  u,x,y,  •••,  by  Au,  /^x,  A?/^  '••  rather  than 
by  du,  dx,  dy,  •••.     Other  notations  are  also  used ;  e.g.  Su,  8x,  Sy,  •••. 

EXAMPLES. 

1.  In  the  cylinder  in  Ex.  3,  Art.  65,  give  an  approximate  value  of  the 
error  in  the  computed  volume  due  to  errors  A^  in  the  height  and  Ar  in 
the  radius. 

Let  F  denote  the  volume.     Then  V=  irr^h. 

.-.  AF  =  2  irrh  ■  Ar  +  tv^  •  Ah. 

The  relative  error  is      ^  =  ?-^  +  M . 
V         r         h 

2.  Do  as  in  Ex.  1  for  a  few  concrete  cases,  and  compare  the  above 
approximate  value  of  the  error  with  the  actual  error.  What  is  the  difference 
between  the  actual  error  in  the  volume  in  Ex.  1  and  its  approximate  value 
obtained  by  the  method  above  ? 

3.  In  the  triangle  in  Ex.  7,  Art.  65,  let  Aa,  A6,  AC,  be  small  errors 
made  in  the  measurement  of  a,  b,  C:  show  that  the  approximate  relative 

error  for  the  computed  area  ^  is  —  +  —  -|-  cot  (7  •  AC. 

a       b 


83,  84.]  IMPLICIT    FUNCTIONS,  139 

Find,  by  the  calculus,  an  approximate  value  of  A  J,  given  that  a  =  20  inches, 
6  =  35  inches,  C  =  48°  30',  Aa  =  .2  inch,  Ab  =  .1  inch,  AC  =  20'.  How  can 
the  actual  error  in  the  computed  area  be  obtained  ? 

4.  Show  that  for  the  area  A  of  an  ellipse  when  small  errors  are  made 

in  the  semiaxes  a  and  6,  approximately  —  =  —  H 

A        a        b 

In  this  general  case,  and  in  several  concrete  cases,  compare  the  approxi- 
mate error  in  the  computed  area  with  the  actual  error. 

5.  In  the  case  described  in  Ex.  3  show  that  if  Ac  denote  the  consequent 
error  in  the  computed  value  of  c,  then,  approximately, 

Ac  =  cos  B  ■  Aa  -\-  cos  A  •  Ab  +  a  sin  B  -  AC. 

N.B.  For  remarks  and  examples  on  this  topic  see  Lamb,  Calculus, 
pp.  138-142,  Gibson,  Calculus,  pp.  258-260. 

84.  Differentiation  of  implicit  functions,  two  variables.  This 
topic  has  been  taken  up  in  one  way  in  Art.  56.  Let  the  relation 
connecting  two  variables  x  and  y  be  in  the  implicit  form 

/(^,  y)  =  c,  ■  (1) 

in  which  c  denotes  any  constant,  including  zero.    Let  u  denote  the 
function  f(x,  y)  ;  then  (1)  may  be  written 

u  =  c.  (2) 

Since  u  remains  constant  when  x  and  y  change,  —  =  0 ;  i.e. 
(Art.  81,  Eq.  3,  and  Note  1) 

dudx     du  dy  _  ^  /o\ 

dx  dt      dy  dt 
dy  du  Q^ 

^^^^  (^)'  I = - 1'  "^"^^^  ^^''- '''  ^^-  ^^)]'  ^ = -  i'  ^^) 

dt         dy  dy 

Ex.  1.    Express  relation  (4)  in  words. 

Note.  It  should  not  be  forgotten  that  the  relation  between  the  function 
and  the  variable  should  be  expressed  in  form  (1)  before  (4)  is  applied. 

Ex.  2.  Do  Exs.  13,  14,  Art.  37,  and  exercises,  Art.  56,  by  the  method  of 
this  article.     Compare  the  methods  of  Arts.  37,  56,  and  84. 


140  INFINITESIMAL   CALCULUS.  [Ch.  Vlll. 

85.  Order  of  partial  differentiations  commutative.  The  theorem 
stated  in  Art.  80,  Note  3,  and  illustrated  by  the  exercises  there,  will  now  be 
proved.     (See  Gibson,  Calculus^  §  93,  especially  pages  221,  222.)     Suppose 

that 

u=f{x,y)  (1) 

and  that  u  and  its  first  and  second  partial  derivatives  are  continuous  over  a 
finite  range  of  the  variables  ;  then,  as  will  now  be  shown, 

dy  doc     doc  dy 

Let  X  receive  an  increment  h^  and  y  remain  constant ;  then  by  the  theorem 
of  mean  value  (Art.  64,  Eq.  3) 

f{x  +  /I,  y)  -  f(x,  y)  =  h  ^-f(x  +  dih,  y),  in  which  0  <  ^i  <  1.  (3) 

Now  let  y  receive  an  increment  k,  and  x  remain  constant ;  then  on  applying 
the  theorem  of  mean  value  to  the  second  member  of  (3), 

[fix  +  h,y^  k)  -fix,  ?/  +  ^•)]  -  [fix  +  h,  y)  -fix,  ?/)] 

=  k^-  [h4-f(x  -f  dih,  y  +  ^2^0l  =  hk-^  l-f /(x  +  M,  y  +  e2k)\      (4)    . 
dyl   dx  J  dyldx  J 

in  which  0  <  ^2  <  1-     On  giving  the  increments  in  the  reverse  order, 

[/(x  -^h,y-\-k)-  fix  -\-h,y)^-  [fix,  y  +  A:)  -  f(x,  y)l 

=  hk-^  rf  fix  +  dsh,  y  +  d,k)\      (5) 

dxldy  J 

in  which  63  and  0^  lie  between  0  and  1.     Hence,  on  equating  the  values  of 

fix  +  fe,  y  +  k)  -fix  +  h,  y)  -fix,  y  +  k)  -\-fix,  y) 
hk 
derived  from  equations  (4)  and  (5), 

V-  -f  /(^  +  ^1^'  y  +  ^2^0  =  ^  ^  fix  +  Bsh,  y  +  ^4^),  (6) 

dy  dx  dx  dy 

provided  that  x  -\-  h  and  y  -\-  k  are  within  the  range  referred  to  above.  Now 
let  h  and  k  approach  zero.  Then,  since  the  first  and  second  partial  deriva- 
tives are  continuous,  equation  (6)  becomes 

d^fjx,  y)  ^  d^fix,  y)  .    .^      d^u   ^    d^u 
dy  dx  dxdy    '       *   dy  dx     dxdy 

The  proof  of  the  commutation  theorem  can  be  extended  to  derivatives  of 
higher  orders  and  to  functions  of  more  than  two  variables. 


85,80.]        CONDITION    FOR    TOTAL    DIFFERENTIAL.  141 

Note  1.  Keferences  for  collateral  reading  on  partial  differentiation, 
total  differentials,  and  the  commutative  property;  Todhunter,  Diff.  Cat.; 
McMahon  and  Snyder,  Diff.  Cal,  Arts.  91-102  ;  Lamb,  Calculus  (ed.  1897), 
Arts.  45,  46,  60-62,  209,  210;  Edwards,  Treatise  on  Diff.  Cal,  Chap.  VI. 
Especially  full  and  clear  treatment  of  differentiation  of  functions  of  more 
than  one  variable,  with  various  illustrations  and  geometrical  interpretation, 
is  given  in  Gibson's  Calculus,  Chap.  XI.  (see  in  particular  pp.  204-225  and 
Ex.  1,  p.  222),  and  in  Echols'  Calculus,  Chaps.  XXV. -XXIX.,  pp.  282-313. 

Note  2.  Applications  of  partial  differentiation :  («)  To  the  determi- 
nation of  the  maximum  and  minimum  values  of  functions  of  tw^o  or  more 
variables  (see  references  in  Art.  76,  Note  7);  (6)  To  the  study  of  surfaces, 
and  curves  in  space  (see  references,  Art.  166,  Note  2). 

86.  Condition  that  an  expression  of  the  form  Pdx  -\-  Qdy  be  a  total 
differential.  This  article  may  be  regarded-  as  supplementary  to 
Art.  82. 

Suppose  that  /^{x,  y)  and  f^ix,  y)  are  two  arbitrarily  chosen 
functions :  does  a  function  exist  which  has  J\(x,  y)  for  its  partial 
a>derivative  and /2(a;,  y)  for  its  partial  .y-deri  vative  ?  A  little 
thought  leads  to  the  conclusion  that  in  general  such  a  function  does 
not  exist.  The  condition  that  must  be  satisfied  in  order  that  there 
may  be  such  a  function  will  now  be  found.  Suppose  that  there  is 
such  a  function,  and  let  it  be  denoted  by  u.  Then,  according  to 
the  hypothesis, 

^=/i(a?,  2/)    and    -~=f.lx,y).  (1) 

By  Art.  85,  _^=-^.  '  (2) 

oy  ox      axoy 

Hence,  from  (1)  and  (2), 

Kesult  (3)  is  directly  applicable  to  the  differential  expression 
Pdx  +  Qdy  on  substituting  P  for  f^(x,  y)  and  Q  for  f2(x,  y). 
Otherwise :  If  Pdx  -f-  Qdy  is  a  total  differential,  du  say,  then 

f'  =  P   and    fi  =  Q.  (4) 

OX  dy 

Hence,  from  (2)  and  (4),     ^=^Q.  (5) 


142  INFINITESIMAL    CALCULUS.  [Ch.VIII. 

When  condition  (5)  is  satisfied,  Pdx  -f-  Qdy  is  also  called  an 
exact  differential. 

Note  1.  That  this  condition  is  not  only  necessary  (as  shown  above),  but 
also  sufficient^  is  shown  in  works  on  Differential  Equations.  {E.g.  see 
Professor  McMahon's  proof  in  Murray,  Diff.  Eqs.,  Note  E.) 

Note  2.  For  the  condition  that  an  expression  of  the  form  Pdx  +  Qdy 
+  Bdz  (see  Art,  82,  Eq.  2)  be  a  total  differential,  see  works  on  Differential 
Equations;  e.g.  Murray,  Diff.  Eqs.,  Art.  102  and  Art.  103,  Note. 

Ex.    1.     Apply   test   (5)    in    the    following  cases  :    (a)    w  =  3  x^  +  2  ?/2  ; 

y 

(6)  u  —  tan  -  ;    (c)  x  dy  -\-  y  dx  ;    (d)  xdy  —  y  dx. 

Ex.  2.  Illustrate  by  examples  the  phrase,  '■'•  in  general  such  a  function 
does  not  exist,"  which  occurs  in  this  article. 

87.    Euler's   theorem   on   homogeneous   functions.      Let  u  be  a 

homogeneous  function  of  x  and  y  of  degree  n ;  i.e.  let 

u  =  Ax""  H h  BxPy''  +  Cxy  -\ \-  My", 

in  which  j9-[-g  =  7'-f-s=  •••  =  n. 

Then        ^  =  uAx""-^  -\ f- pBx^-y  +  rCx'-^y'  -\ ; 

ax 

-^  =  •••  4-  qBx^r^  4-  sCxY~^  H h  nMy'^'K 

dy 

From  this,  on  multiplication  and  simplification, 

doc         dy  ^  ^ 

This  result  can  be  extended  to  homogeneous  functions  of  any 
number  of  variables  ;  thus, 

^|^+2/^+«^  +  ...  =  nt«.  (2) 

due         dy         dz 

Result  (2)  is  called  Euler's  theorem.* 

(See  Williamson,  Diff.  Cal.,  Arts.  102-104,  123 ;  McMahon  and  Snyder, 
Diff.  Cal,  Art.  100 ;  Gibson,  Calculus,  page  412.) 

*  From  Leonhard  Euler  (1707-1783),  an  eminent  Swiss  mathematician,  who 
worked  at  Berlin  and  St.  Petersburg.  He  greatly  advanced  the  subjects  of 
algebra,  trigonometry,  and  the  calculus. 


87,88.]  SUCCESSIVE  TOTAL  DERIVATIVES.  143 

Ex.  1.   Prove  theorem  (2)  when  u  is  a  homogeneous  function  of  oj,  y,  z. 
Ex.  2.    Illustrate  (1)  and  (2)  by  examples  in  which  n  is  an  integer. 
Ex.  3.    Verify  Euler's  thieorem  in  the  following  cases  : 

(i)  w  =  (x^  +  ?/3)  (x*  +  y^)  ;     (ii)   u  =  {x^  +  vh  ^  iS^  +  vh  \ 

(iii)  if  =  sin-i  ^^  ~  ^^     (Here  ?i  =  0.) 

.  Vx  +  Vy 

Ex.  4.    Verify  Euler's  theorem  when  u—f\^-\\    and  apply  to  tan ^, 
sin~i  ^,  log^,  in  particular.     (In  this  /-function  n  =  0.) 

X  X 

88.    Successive  total  derivatives.     An  example  will  be  given  in  order 
to  show  the  procedure. 

If  w=/(x,  2/),  (1) 

then  (Art.  81,  Eq.  3)  du^dndx     dudy^  ,2) 

dt      dx  dt      dy  dt  ^  ^ 

On  differentiation  with  respect  to  t  again, 

dt^  ~  dt\dtj~  dt\dxl  '  dt      dx  dt^     dt\dy  )  '  dt      dy  dt^'       ^'  ^ 

Now,  in  general,  ^  and  ^  are  functions  of  x  and  y  ;  hence,  on  applying 
dx  dy 

the  principle  enunciated  in  Art.  81, 

djdu\d_(di(\  ,dx_^d_(du\  ,  dy 
dt\dx)     dx\dx)  '  dt      dy\dx)     dt' 

dt\dy)    dx\dy)    dt     dy\dy)    dt 

On  substituting  these  values  in  (3),  using  the  notation  of  Art.  80,  and 

remembering  that    o  ^   —  ^  ^^  ,  Equation  (3)  becomes 
dxdy     dydx 

^^d^fdxy     ^_^dxdy     d^(dyy     du^,  /4>) 

dt^      dx^dtl  dxdy  dt  dt      dyAdt)       dx  dt^      dy  dt^ 

N.B.     Questions  and  exercises  suitable  for  practice  and  review  on  the 
subject-matter  of  this  chapter  will  be  found  at  page  386 


CHAPTER   IX. 

CHANGE    OF    VARIABLE. 

N.B.  If  it  is  thought  desirable,  the  study  of  this  chapter  may  be  post- 
poned until  some  of  the  following  chapters  are  read. 

89.  Change  of  variable.  It  is  sometimes  advisable  to  change 
either,  or  both,  of  the  variables  in  a  derivative.  If  the  relation 
between  the  old  and  the  new  variables  is  known,  the  given 
derivative  can  be  expressed  in  terms  of  derivatives  involving  the 
new  variable,  or  variables.  Arts.  91-93  are  concerned  with 
showing  how  this  may  be  done.  In  Art.  90  an  expression  for  the 
given  derivative  is  found  when  the  dependent  and  independent 
variables  are  interchanged ;  in  Art.  91,  when  the  dependent 
variable  is  changed ;  in  Art.  92,  when  the  independent  variable 
is  changed ;  and  in  Art.  93,  when  both  the  dependent  and  the 
independent  variables  are  expressed  in  terms  of  a  single  new 
variable.  In  Note  1,  Art.  93,  an  example  is  worked  in  which  the 
dependent  and  the  independent  variables  are  both  expressed  in 
terms  of  two  new  variables. 

N.B.     Principle  (2)  of  Art.  34  is  repeatedly  employed  in  Arts.  90-93. 

90.  Interchange  of  the  dependent  and  independent  variables.  Let 
y  be  the  dependent  and  x  the  independent  variable.  This  article 
shows  how  to  express  the  successive  derivatives  of  y  with  respect 
to  X  in  terms  of  the  derivatives  of  x  with  respect  to  y. 

From  the  fact  that  ^  .  ^  =  1,  and  Art.  20  (c),  it  follows  that 


Ay~ 

■1,  ai 

dy 
dx 

1 
~d.x 

dy 
144 

89-92.] 

CHANGE  OF   VARIABLE. 

Again, 

d'y_  d  (dy\_  d  fdy\     dy     ...    o^x 
dor'  -  dx[dx)  -  dy[dx)  '  dx     ^^'^^  ^> 

d 

r  1  1 

d^x 
dx           dy^ 

'^dy 

dx 

dy. 

'  dy~      fdxV 

\dyj 

Ex.   Express 

the  third  aj-derivative  of  y  in  terms  of  ^-derivatives  of  x. 

145 


91.  Change  of  the  dependent  variable.  Let  the  dependent  and 
independent  variables  be  denoted  by  y  and  x  respectively.  It  is 
required  to  express  the  successive  derivatives  of  y  with  respect  to 
x,  in  terms  of  the  derivatives  of  z  with  respect  to  x  when 

y  =  F(z). 


dy^dyd^^^dz^^ 
dx     dz    dx         ^  ^    dx' 


^^d^/dy 
dx^     dx  \dx 


=  lh)|] 


^Ux'     dx   dx     ^^         ^^d^     dx   dz^    ^^-'    dx 


Ex.    Given  that  y  =  F(z) ,  show  that 

dx^         ^  ^  dx^  ^  ^    dx^dx  ^  ^  \dx) 

92.  Change  of  the  independent  variable.  Let  the  dependent  and 
independent  variables  be  denoted  by  y  and  x  respectively.  It  is 
required  to  express  the  successive  derivatives  of  y  with  respect  to 
ic,  in  terms  of  the  derivatives  of  y  with  respect  to  z  when 


x=f(z). 
ad  hence, 

dy  _dy     dz  __     1        dy 
dx~ dz     dx~  f'(z)     dz 


!=/(.),  and  hence,  1  =  ^ 


146 


INFINITESIMAL    CAL CULUS. 


[Ch.  IX. 


d^y  _  d^  (dy\  _  d  fdy\     dz  _d_/'    1     dy\     dz_ 
dx^     dx  \dxj      dz  \dxj     dx     dz  \f'(z)  dzj     dx 


1        d'y       f"(z)      dy 


f{z)y\z)     dz'      \_r(z)J    dz] 


d^y 


Ex.    Find  -r-|  when  x—f{z). 


93.   Dependent  and  independent  variables  both  expressed  in  terms 
of  a  single  variable. 

Let  yz=z(fi(t)  and  x=f(t). 


Then 


dy_dy  ^  dx  /oxn  _  <^'(0 


d^y  _  d  fdy\  _  d  fdy  \     dt  _d 


\^(^(dy\ 
J      dt  \dxj 


dx-     dx  \dxj      dt  \dxj     dx     dt 

_f{t)^"{t)-<i>xt)r(t) 

[.f'it)7 


1 

fit) 


(Compare  Art.  71.) 


EXAMPLES. 

1.  In  the  above  case  find  — |- 

2.  Given  that  x  =  a(d  —  sin  6)  and  y  =  a(l  —  cos  6),  calculate 

b  +  {l^yf^%-  (See  EX.  9,  An.  68.) 

3.  Given  that  x  =  a  cos  6  and  y  =  asm  d,  calculate  the  same  function 
as  in  Ex.  2.     What  curve  is  denoted  by  these  equations  ? 

4.  Given  that  x  =  acosd  and  y  =  bsm  d,  calculate  the  same  function  as 
in  Ex.  2.     What  curve  is  denoted  by  these  equations  ? 

Note  1.     Both  dependent  and  independent  variables  expressed  in  terms  of 
two  new  variables.     Following  is  an  example  of  this. 

Ex.   Given  the  transformation  from  rectangular  to  polar  coordinates,  viz. 

aj=:rcos^,   ?/=:rsin^,  (1) 

express  ^  and  — ^  in  terms  of  r,  ^,  and  the  derivatives  of  r  with  respect  to  6. 


93.]  CHANGE  OF  VARIABLE.  147 

From  (1),     —  =  cose~-rsme,    ^  =  sin^— +  r cos^. 
^  ^      dd  dd  dd  dd 


•••i=(i4:'---.-^(^)) 


dr 

sm  ^  ~  4-  r  cos  d 


dr 
cosd  —  —  r  sin  8 
dd 


d^_  d  fdy\_d  fdy\     dd  _  \dd  J         dff^ 

d^-^W-d^Uy'^~/cos^|-rsin^V' 

Note  2.  For  more  complex  cases  of  change  of  the  variables  in  a  derivative, 
see  other  text-books. 

Note  3.  References  for  collateral  reading.  Williamson,  Diff.  Cal., 
Chap.  XXII.  ;  McMahon  and  Snyder,  Diff.  CaL,  Chap.  XI.;  Edwards, 
Treatise  on  Diff.  Cal,  Chap.  XIX.  ;   Gibson,   Calculus,  §§  98,  99. 


EXAMPLES. 

N.B.  In  working  these  examples  it  is  much  better  not  to  use  the  results 
or  formulas  derived  in  Arts.  90-93,  but  to  employ  the  method  by  which  these 
results  have  been  obtained. 

1.  Change  the  independent  variable  from  x  to  y  in  :  (i)  — ^  +2y[-^j  =0; 

(ii)  s(m-^-iy^-fyl^y=o.  ^""^     ^^""^ 

^  ^     [djc^l      dxdx^      dxAdxj 

2.  In  ^  =  1  +  ^^^  +  y^  f  ^  Y,  change  the  dependent  variable  from  y  to 

dx-  1  +  y^    \dx  I 

0,  given  that  y  =  tan  z. 

Z^    Change   the   independent   variable   under   the    following    conditions : 

niWf^+x^  +  u  =  0,y  =  \ogx;iu)(l-x'^)'^-xf--  +  y=0,x=:cost-, 
^^-^     dx^         dx  dx^         dx 

{m)(l-x^)^-x^  =  0,x  =  cost;    {iv)  x'^fy  +  2x^ -{-^y  =  0,  xz  =  1; 
dx^        dx       vv  A  ^  ^  *-  ^  J,  dx^  dx     x^ 

^Sx^  +  y  =  \ogx,x  =  e'. 
dx 

4.  Find  ^  and  ^  when  :  (i)x  =  a  (cos  t -^tsint),  y  =  a  (sin  t-tcost); 

dx  dx^ 

(ii)  X  =  cot  t,  y  =  sin-^  t. 

5.  If  r.^-?f^Y+^  =  0,  andx  =  ye^showthat2/^  +  ^  =  0. 

dx^     y\dx)      rfx       '  ^    '  dy^     dy 


CHAPTER   X. 

INTEGRATION. 

N.B.  If  thought  desirable,  Art.  97  may  be  studied  before  Arts.  95,  96. 
(Remarks  relating  to  the  order  of  study  are  in  the  preface.) 

94.  Integration  and  integral  defined.  Notation.  In  Chapter  III. 
a  fundamental  process  of  the  calculus,  namely,  differentiation^ 
was  explained.  In  this  chapter  two  other  fundamental  processes 
of  the  calculus,  each  called  integration,  are  discussed.  The 
process  of  differentiation  is  used  for  finding  derivatives  and 
differentials  of  functions ;  that  is,  for  obtaining  from  a  function, 
say  F{x),  its  derivative  F'(x),  and  its  differential  F'{x)dx.  On 
the  other  hand  the  process  of  integration  is  used : 

(a)  For  finding  the  limit  of  the  sum  of  an  infinite  number  of 
infinitesimals  which  are  in  the  differential  formf(x)dx  (see  Art.  96) ; 

(h)  For  finding  functions  whose  derivatives  or  differentials  are 
given  ;  that  is,  for  finding  anti-derivatives  and  anti-differentials 
(see  Arts.  27  a,  97). 

Briefly,  integration  may  be  either  (a)  a  process  of  summation, 
or  (h)  a  jy^'ocess  which  is  the  inverse  of  differentiation,  and  which, 
accordingly,  may  be  called  ariti-differentiation.  Integration,  as  a 
process  of  summation,  was  invented  before  differentiation.  It 
arose  out  of  the  endeavor  to  calculate  plane  areas  bounded  by 
curves.  An  area  was  (supposed  to  be)  divided  into  infinitesimal 
strips,  and  the  limit  of  the  sum  of  these  was  found.  The  result 
was  the  ivhole  (area) ;  accordingly  it  received  the  name  integral, 
and  the  process  of  finding  it  was  called  integration.  In  many 
practical  applications  integration  is  used  for  purposes  of  sum- 
mation. In  many  other  practical  applications  it  is  not  a  sum 
but  an  anti-differential  that  is  required.  It  will  be  seen  in  Art.  96 
that  a  knowledge  of  anti-differentiation  is  exceedingly  useful  in 
the  process  of  summation.  Exercises  on  anti-differentiation  have 
appeared  in  preceding  articles. 

148 


94.  j  INTEGRATION.  149 

Note.  The  part  of  the  calculus  which  deals  with  differentiation  and  its  im- 
mediate applications  is  usually  called  The  Differential  Calculus^  and  the  part 
of  the  calculus  which  deals  with  integration  is  called  The  Integral  Calculus. 
With  Leibnitz  (1646-1716),  the  differential  calculus  originated  in  the  problem 
of  constructing  the  tangent  at  any  point  of  a  curve  whose  equation  is  given. 
This  problem  and  its  inverse,  namely,  the  problem  of  determining  a  curve 
when  the  slope  of  its  tangent  at  any  point  is  known,  and  also  the  problem  of 
determining  the  areas  of  curves,  are  discussed  by  Leibnitz  in  manuscripts 
written  in  1673  and  subsequent  years.  He  first  published  the  principles  of 
the  calculus,  using  the  notation  still  employed,  in  the  periodical.  Acta 
Eruditorum,  at  Leipzig  in  1684,  in  a  paper  entitled  Nova  methodus  pro 
maximis  et  minimis,  itemque  tangentibus,  quae  nee  fractas  nee  irrationales 
quantitates  moratur,  et  singulare  pro  illis  calculi  genus.  Isaac  Newton 
(1642-1727)  was  led  to  the  invention  of  the  same  calculus  by  the  study  of 
problems  in  mechanics  and  in  the  areas  of  curves.  He  gives  some  description 
of  his  method  in  his  correspondence  from  1669  to  1672.  His  treatise, 
Jlethodus  fluxionum  et  serierum  infinitarum,  cum  ejusdem  applicatione  ad 
curvarum  geometriam,  was  written  in  1671,  but  was  not  published  until  1736. 
The  principles  of  his  calculus  were  first  published  in  1687  in  his  Principia 
{Fhilosophiae  Naturalis  Principia  Mathematical.  It  is  now  generally 
agreed  that  Newton  and  Leibnitz  invented  the  calculus  independently  of  each 
other.  For  an  account  of  the  invention  of  the  calculus  by  Newton  and 
Leibnitz,  see  Cajori,  History  of  Mathematics,  pp.  199-236,  and  Cantor, 
Geschichte  der  Mathematik,  Vol.  3,  pp.  150-172. 

"  There  are  certain  focal  points  in  history  toward  which  the  lines  of  past 
progress  converge,  and  from  which  radiate  the  advances  of  the  future.  Such 
was  the  age  of  Newton  and  Leibnitz  in  the  history  of  mathematics.  During 
fifty  years  preceding  this  era  several  of  the  brightest  and  acutest  mathe- 
maticians bent  the  force  of  their  genius  in  a  direction  which  finally  led  to  the 
discovery  of  the  infinitesimal  calculus  by  Newton  and  Leibnitz.  Cavalieri, 
Roberval,  Fermat,  Descartes,  Wallis,  and  others,  had  each  contributed  to 
the  new  geometry.  So  great  was  the  advance  made,  and  so  near  was  their 
approach  toward  the  invention  of  the  infinitesimal  analysis,  that  both 
Lagrange  and  Laplace  pronounced  their  countryman,  Fermat,  to  be  the  true 
inventor  of  it.  The  differential  calculus,  therefore,  was  not  so  much  an 
individual  discovery  as  the  grand  result  of  a  succession  of  discoveries  by 
different  minds."     (Cajori,  History  of  Mathematics,  p.  200.) 

Also  see  the  "Historical  Introduction"  in  the  article,  Infinitesimal  Cal- 
culus {Ency.  Brit.,  9th  edition),  and,  at  the  end  of  that  article,  the  list  of 
works  bearing  on  the  infinitesimal  method  before  the  invention  of  the 
calculus. 

Notation.  In  differentiation  d  and  D  are  used  as  symbols  ;  thus, 
df{x)  is  read  "  the  differential  of  /(»),"  and  Df(x)  is  read  "  the 


150 


INFINITESIMA L   CALCUL US. 


[Ch.  X. 


derivative  of  /(a?)."  In  integration,  whether  the  object  be  sum- 
mation or  anti-differentiation,  the  sign  |  is  most  generally  used 
as  the  symbol ;   thus,    |  f(x)  dx  is  read  "  the  integral  off(x)  dx^  * 

Other  symbols,  viz.  d'~^f{x)dx  and  D~'^f(x),  are  used  occasionally 
(see  Art.  97,  Xote  2).  The  quantity  f(x)  which  appears  "under 
the  integration  sign,"  as  the  mathematical  phrase  goes,  is  called 
the  integrand. 

95.  Examples  of  the  summation  of  infinitesimals.  These  examples 
are  given  in  order  to  help  the  student  to  understand  clearly  what 
the  phrase  "  to  find  the  limit  of  the  sum  of  a  set  of  infinitesimals 
of  the  ioTiR  f(x)dx  {i.e.  a  set  of  infinitesimal  differentials)"  means. 

(a)  Find  the  area  between  the  line  y  =  mx,  the  x-axis,  and  the  ordinates 

drawn,  to  the  line  at 
X  =  a  and  x  =  b. 

Let  PQ  be  the  line 
whose  equation  is 
y  =  mx,  OA  =  a,  and 
OB  =  b.  Draw  the 
ordinates  ^Pand  BQ ; 
it  is  required  to  find 
the  area  APQB. 

Suppose  that  AB 
is  divided  into  n  equal 
parts  each  equal  to  Ax, 
X    so  that 

n  •  Ax  =  b  —  a. 


r 

Pj 

P 

■P«-l 

? 
^ 

X-^ 

p, 

y 

G 

s                                  -S 

0 

^ 

L   Jl 

h^ 

M.                                   Mn 

-1^ 

3        3 

Fig.  39. 


Draw  the  ordinates  at  each  point  of  division,  M^  M2,  •••,  il/n-i ;  complete 
the  inner  rectangles  PMi,  Pi,  ^¥2,  •■•,  Pn-iB  ;  and  complete  the  outer  rectan- 
gles PiA,  P2M1,  •••,  QMn-i.  The  area  APQB  is  evidently  greater  than  the 
sum  of  the  inner  rectangles  and  less  than  the  sum  of  the  outer  rectangles  ;  i.e. 

sum  of  inner  rectangles  <  APQB  <  sum  of  outer  rectangles. 


*  The  word  integral  appeared  first  in  a  solution  of  James  Bernoulli  (1654- 
1705),  which  was  first  published  in  the  Acta  Eruditornm  in  1690.  Leibnitz 
had  called  the  integral  calculus  calculus  .ncmmatorius,  but  in  1696  the  term 
calculus  integralis  was  agreed  upon  by  Leibnitz  and  John  Bernoulli  (1667- 
1748).  The  sign  \  was  first  used  in  1675,  and  is  due  to  Leibnitz.  It  is 
merely  the  long  S  which  is  the  initial  letter  of  summa,  and  was  used  by 
earlier  writers  to  denote  "  the  sum  of." 


96.]  *  INTEGRATION.  151 

The  difference  between  the  sum  of  the  inner  and  the  sum  of  the  outer  rectangles 
is  the  sum  of  the  rectangles  PPi,  P1P2,  •••,  P"~^Q-  The  latter  sum  is  evidently 
equal  to  the  rectangle  QS,  i.e.  to  CQ  ■  Ax.  This  approaches  zero  when  Ax 
approaches  zero.  Therefore  APQB  is  the  limit  of  the  sum  of  either  set  of 
rectangles  when  Ax  approaches  zero.  The  limit  of  the  sum  of  the  inner 
rectajigles  will  now  be  found. 


At^, 

X  =  a^ 

and  hence, 

AP  =  ma ; 

atilfi, 

x  =  a-\-  Ax, 

and  hence, 

MiPi  =  m{a  +  Ax)  ; 

at  Jf2, 

x  =  a  +  2Ax, 

and  hence. 

MiPi  -  m(a  +  2  Ax)  ; 

at  Mn-i,  x  =  a  +  n  —  1  Ax,  and  hence,  ilf„_iP„_i  =  7)i(a  +  n  —  1  •  Ax), 

.'.  sum  of  inner  rectangles 

=  ma  •  Ax  +  m(a  +  Ax)  •  Ax  +  m(a  +  2  Ax)  •  Ax  +  ••• 


+  m(a  +  n  —  I  '  Ax)  •  Ax. 
/.  area  APQB  =  lim^^^  [ma  Ax  +  m(a-]-Ax)Ax-{--"-\-m(a-\-n-l  •  Ax) Ax] 

=  limAx=o»i[a+(rt  +  Ax)+(a+2  Ax)H [-{a  +  n  —  1  •  Ax)]Ax. 

Hence,  on  summation  of  the  arithmetic  series  in  brackets, 
mn  Ax , 


area  APQB  =  limAx^      [      {2  a -^  n  -  I  ■  Ax}. 
On  giving  n  Ax  its  value  h  —  a,  this  becomes 
area  APQB  =  limAx-o  ^"^^~^^  (6  +  a  -  Ax) 


-"{1-1} 


Note  1.  In  this  example  the  element  of  area,  as  it  is  called,  is  a  rectangle 
of  height  y  and  width  Ax  when  Ax  is  made  infinitesimal,  i.e.  the  element 
of  area  is  y  dx  or  mx  dx  in  which  dx  =  0.  (See  Art.  27,  Notes  3,  4,  and 
Art.  67rt.) 

Note  2.     It  may  be  observed  in  passing  that  on  taking  the  anti-differential 

of  mxdx,  namely  ^^,  substituting  h  and  a  in  turn  for  x  therein,  and  taking 

the  difference  between  the  results,  the  required  area  is  obtained. 

Ex.  Find  the  limit  of  the  sum  of  the  outer  rectangles  when  Ax  approaches 
zero. 

(&)  Find  the  area  between  the  parabola  y  =  ofi,  the  x-axis,  and  the  ordinates 
atx  =  a  and  x  =  b. 


152 


INFINITESIMA L   CALC UL US. 


[Ch.  X. 


Let  LOQ  be  the   parabola,   OA  =  a,    OB  =  h  ;   draw  the  ordinates  AP 

and  BQ;  the  area  APQB  is 
required.  As  in  the  preceding 
problem,  divide  AB  into  n 
parts  each  equal  to  Aoj,  so  that 

draw  ordinates  at  the  points 
of  division,  and  construct  the 
set  of  inner  rectangles  and 
the  set  of  outer  rectangles. 
As  in  (a),  it  can  be  seen  that 
sum  of  inner  rectangles  < 
area  APQB  <  sum  of  outer  rectangles  ;   and  also  that 

(sum  of  outer  rectangles)  —  (sum  of  inner  rectangles)  =  CQ  •  Asc, 

which  approaches  zero  when  Ax  approaches  zero.  Hence  the  area  APQB  is 
the  limit  of  the  sum  of  either  set  of  rectangles  when  Ax  approaches  zero. 
The  limit  of  the  sum  of  the  inner  rectangles  will  now  be  found. 


At^, 

X  =  a, 

and  hence. 

AP=a'^', 

at  ilfi, 

x  =  a-h  Ax, 

and  hence. 

i¥iPi  =  (a  +  Ax)2; 

atiJfa, 

X  =  a  +  2  Ax, 

and  hence, 

M2P2  =  (a  +  2  Ax)'^ ; 

at  Mn-i,         x  =  a  +  n  —  1  ■  Ax,  and  hence,  Mn-iPn-i  =  (a  +  n  —  1  -  Ax)2. 
.-.  sum  of  inner  rectangles  =  a'^Ax  +  («  +  Ax)2Ax  +  (a  +  2  Ax)2Ax  +  ••• 


+  (a  +  w  -  1  •  Ax)2Ax. 
area  APQB  =  \im^^^{a^  +  («  +  Ax)2  -f  («  +  2  Ax)2  + 


+  (a  +  w  -  1  •  Ax)2}Ax 


=  liraAx^o{na2  +  2  a  Ax(l  +  2  +  3  H h  n  -  1) 


+  (Ax)2(12  +  22  +  32  +  ...  +  w  -  r)}Ax. 


Now 
and 


1  +  2  +  3  +  .-. +  w-l  =  i  n(n  -  1)  ; 


12  4.  22  +  32  +  ...  +  n  -  r  =  i  (n  -  l)n(2  n  -  1).* 
area  APQB  =  limAx:^ n  Ax {a^  +  aw  Ax  —  a  Ax  +  |  (n  Ax)2 
-Jw(Ax)2+KAx)2}. 


*  It  is  shown  in  algebra  that  the  sum  of  the  squares  of  the  first  n  natural 
numbers,  viz.  I2,  22,  32,  ...,  n2,  is  ^  n(w  +  1)  (2  n  +  1). 


95.]  INTEGRATION.  153 

But  w  Aic  =  6  —  a,  no  matter  what  n  and  Ax  may  be. 

.-.  area  APQB  =  limAxio  {h  -  a){a^  -\r  a{h  -  a)  -  a  ^x  ^- \(h  -  a)^ 

3       3" 

Note  1.  In  this  example  the  element  of  area  is  a  rectangle  of  height  y 
and  width  Aoj,  when  Ax  becomes  infinitesimal,  i.e.  the  element  of  area  is 
y  dx^  I.  e.  x^  dx,  in  which  dx  =  0. 

Note  2,  It  may  be  observed  in  passing  that  the  result  (1)  can  be  ob- 
tained by  taking  the  anti-differential  of  x'^  dx,  namely  — ,  substituting  b  and 

a  in  turn  for  x  therein,  and  calculating  the  difference ^• 

3       o 

Ex.   Find  the  limit  of  the  sum  of  outer  rectangles. 

(c)  Find  the  distance  through  which  a  body  falls  from  rest  in  ti  seconds, 
it  being  known  that  the  speed  acquired  in  falling  for  t  seconds  is  gt  feet  per 
second.     [Here  g  represents  a  number  whose  approximate  value  is  32.2.] 

Note  1.  If  the  speed  of  a  body  is  v  feet  per  second  and  the  speed  remains 
uniform,  the  distance  passed  over  in  t  seconds  is  vt  feet. 

Let  the  time  ti  seconds  be  divided  into  n  intervals  each  equal  to  At,  so  that 

nAt  =  ti. 

The  speed  of  the  falling  body  at  the  beginning  of  each  of  these  successive 
intervals  of  time  is 

0,  g  '  At,  2  g  '  At,  •••,  (n  -  l)g  •  At,  respectively  ; 

the  speed  of  the  falling  body  at  the  end  of  each  successive  interval  of  time  is 

g  •  At,  2g  •  At,  S  g  •  At,  •••,  7ig  •  At,  respectively. 

For  any  interval  of  time  the  speed  of  the  falling  body  at  the  beginning  is 
less,  and  the  speed  at  the  end  is  greater,  than  the  speed  at  any  other  moment 
of  the  interval.  Now  let  the  distance  be  computed  which  would  be  passed 
over  by  the  body  if  it  successively  had  the  speeds  at  the  beginnings  of  the 
intervals  ;  and  then  let  the  distance  be  computed  which  would  be  passed  over 
by  the  body  if  it  successively  had  the  speeds  at  the  ends  of  the  intervals. 

The  first  distance      =  0  +  g^Aty^  +  2  g^Aty  -I-  •••  +  (n  -  1)</(A«)2 

=  [0  +  l+2+...+(;i-l)]^(A02 

=  in(n-l)g(Aty. 


154 


INFINITESIMAL   CALCULUS. 


[Ch.  X. 


The  second  distance  =:[l-|'2  +  3+---  +  «]^(A«)2 

The  actual  distance  fallen  through,  which  may  be  denoted  by  s,  evidently 
lies  between  these  two  distances  ^  i.e. 

J  n{n  -  \)g{MY  <  s  <  ^  «(ji  +  1)^(A02. 

On  putting  ti  for  its  equal,  n  Ai,  this  becomes 

^gtr'  -^gh'M<s<\  gh^  +  \gh-  At. 

On  letting  At  approach  zero  these  three  distances  approach  equality,  and 

hence  s  =  ^  gt{^. 

Note  2.     For  two  other  examples  see  Art.  96,  Note  4. 

96.  Integration  as  summation.  The  definite  integral.  It  will 
now  be  shown,  geometrically,  how  integration  is  a  jjrocess  of  sum- 
mation.    Let  f{x)  denote  any  function  of  x  which  is  continuous 

from  x  =  a  to  x  =  h  and  geometri- 
cally representable.  Let  its  graph 
be  the  curve  K  whose  equation  is 

accordingly  ^.  . 

y  =f{^)' 

Suppose  that  OA  =  a  and  OB  =  b, 
and  draw  the  ordinates  AP  and  BQ. 
Divide  AB  into  7i  parts,  each  equal 

to  Ace ;  accordingly. 


Pn-lQ. 


n  Ax  =  b 


(1) 


At  the  points  of  division  erect  ordinates,  and  construct  inner 
and  outer  rectangles  as  in  Art.  95  (a),  (6).  It  can  be  shown,  as 
in  the  examples  in  Art.  95,  that  the  difference  between  the  set  of 
the  inner  rectangles  and  the  set  of  the  outer  rectangles  is  CQ  •  Ax 
(CQ  being  equal  to  BQ  —  AP),  a  difference  which  approaches 
zero  when  Ax  approaches  zero.  The  area  APQB  lies  between 
these  sets  and  evidently  is  the  limit  of  the  sum  of  either  set  of 
rectangles  when  Ax  approaches  zero.  The  limit  of  the  sum  of 
inner  rectangles  will  now  be  found. 


96.]  .      INTEGRATION.  155 

At  ^,       x  =  a,  and  hence,  AP  =f(a)  ; 

at  Jii,  x  =  a-}-  Ax,     and  hence,         M^P^  =  f(a  +  Ax)  ; 

at  Jfg,  X-  =  a  H-  2  Aa.-,  and  hence,         M.2P2  =f{ci  +  2  Aa;)  ; 

at  Jfn-i,        a;  =  Z>  —  Ao;,     and  hence,  iJf„_iP„_i  =/(^  —  Ax). 

.:  area  APQB  =  lini_^^^o 
lf(a)Ax-{-f(a-{-Ax)Ax-{-f{a-^2Ax)Ax+"'  -\-f(b-Ax)Axl.  (2) 

The  second  member,  which  is  the  sum  of  the  values,  infinite  in 
number,  that  f(x)Ax  takes  when  x  increases  from  a  to  b  by  equal 
infinitesimal  increments  Ax,  may  be  written  (i.e.  denoted  by) 

\im:,,^Q^f(x)Ax.* 

It  is  the  custom,  however,  to  denote  the  second  member  of  (2) 
by  putting  the  old-fashioned  long  *S'  before  f{x)clx  and  writing  at 
the  bottom  and  top  of  the  *S'  respectively  the  values  of  x  at  which 
the  summation  begins  and  ends ;  thus 

f(3c)dic;  or,  more  briefly,    I    f{x)dQC,  (3) 

This  symbol  is  read  "the  integral  of  f(;x)clx  between  the  limits 
a  and  b"  or  "  the  integral  of  f(x)clx  from  a;  =  a  to  a;  =  6." 

Note  1.  The  numbers  a  and  b  are  usually  called  the  Iqicer  and  upper 
limits  of  X.  It  would  be  better,  perhaps,  not  to  use  the  word  limit  in  this 
connection,  but  to  say  "the  initial  and  final  values  of  x,"  or  simply,  "the 
end-values  of  a:."  f 

Note  2.  The  infinitesimal  differential  f(x)clx  is  called  "ff/i  element  of 
the  integral.  It  is  the  area  of  an  infinitesimal  rectangle  of  altitude  f{x)  and 
infinitesimal  base  dx. 


*  The  latter  part  of  this  symbol  denotes,  and  is  to  be  read,  "the  sum  of 
all  quantities  of  the  type"  [or  "form"]  "/(x)A:t,  from  x  =  a  to  x  =  6" 
[or  "  between  x  =  a  and  x  — &"]. 

t  Joseph  Fourier  (1768-1830)  first  devised  the  way  shown  in  (3)  of  indi- 
cating the  end-values  of  a:. 


156  INFINITESIMAL    CALCULUS.  [Ch.  X. 

Note  3.  It  is  not  necessary  that  the  infinitesimal  bases,  i.e.  the  increments 
Ax  of  X,  toe  all  equal ;  but  for  purposes  of  elementary  explanation  it  is  some- 
what simpler  to  take  them  as  all  equal.  (See  Lamb,  Calculus^  Arts.  86,  87, 
and  the  references  in  Art,  97,  Note  o  ;  also  Snyder  and  Hutchinson,  Calculus, 
Art.  150.) 

Note  4.     For  the  calculation  of    \    eHlx  and   i    s'mxdx  by  the  process 

shown  in  Art.  95,  see  Echols,  Calculus,  Art.  125. 

The  sum  in  brackets  in  (2)  will  now  be  calculated,  and  then  its 
limit,  which  is  indicated  by  the  symbol  (3),  will  be  found. 

Let  the  anti-differential  (A  rt.  27  a)  of  f(x)  dx  *  be  denoted  by 
<^(x);thatis,let  f(,)ax^a.i>{x). 

Then,  by  the  elementary  principle  of  differentiation  (see  Art.  22, 
Note  3)  for  all  values  of  x  from  a  to  6, 

i(^±^|^iiM  =/(»)  +  «,  (4) 

in  which  e  denotes  a  function  whose  value  varies  with  the  value 
of  X,  and  which  approaches  zero  when  Aa;  approaches  zero.  On 
clearing  of  fractions  and  transposing,  (4)  becomes 

f{x)  ^x  =  <f>{x-{-  A.t)  —  <^  (.t)  —  e  •  Ax.  (5) 

On  substituting  a,  a  -f  Aa.',  a  -i-2  Ax,  •••,  h  —  Ax  in  turn  for  x  in 
(5),  and  denoting  the  corresponding  values  of  e  by  e^,  e^,  e.^,  ••♦,  e„, 
respectively,  there  is  obtained : 

f(a)  /\x  =  <f}(a-\-     Ax)  —  <^  (a)  —  e^  •  Ax, 

/(a  +  Ax)  Ax  =  <^  (a  -f  2  Ax)  —  </>  (a  -|-    Ax)  —  e.2  •  Ax, 

f(a  +  2  Ax)  Ax  =  ^  (a  +  3  Ax)  —  <^  (a  +  2  Ax)  —  e^  •  Ax, 


f(b-Ax)Ax=<f>(b)  -<^(6-Ax)     -e„.Ax. 

*  If  f(x)  is  a  continuous  function  of  oj,  /(x)  dx  has  an  anti-differential.  For 
proof  see  Picard,  Traite  cf  Analyse,  t.  I.  No.  4 ;  also  see  Echols,  Calculus, 
Appendix,  Note  9. 


96.]  INTEGRATION.  157 

Addition  gives 

/(a)  Ax  -\-f(ci  4-  Ax)  Ax  +/(a  +  2  Ax)  Ax  H \-f{h  -  Ax) 

=  4>(h)-  <\>(a)  -  (ei  +  e,  +  ^3  +  ...  +  e„)  Ax.  (6) 

Oil  taking  the  limit  of  each  member  of  (6)  when  Ax  approaches 
zero, 

jj{x)  dx  =  cf>(b)-cf>  (a)  -  lim^^o  (e^  4-  e,  -f- . . .  4-  e,)  Ax.       (7) 

Let  ej  be  one  of  the  e's  which  has  an  absolute  value  E  not  less 
than  any  of  the  others  ;  then  evidently 

(ei  -h  ^2  H +  e„)  Ax  <  iiE^x ; 

i.e.  by  (1),  (e^  +  ^^  +  •  •  •  +  e„)  Ax  <  (b  -  a)  E. 

Hence,  lim^^^  (^i  +  ^9  H-  •  •  •  +  e„)  Ax  =  0,  since  E  approaches  zero 
when  Ax  approaches  zero ;  and  therefore, 

JV(a?)  dx  =  <|>(6)  -  c|>(a).  (8) 

That  is,  expressing  (8)  in  words  :   The  integral   I  f(x)  dx,  which 

•  /a 

is  the  limit  of  the  sum  of  all  the  values,  infinite  in  number,  that 
f(x)  dx  takes  as  x  varies  by  infinitesimal  increments  from  a  to  b,  is 
obtained  by  finding  the  anti-differential,  cf>(x),  off(x)dx,  and  then 
calculating  <f)(b)  —  <t>{a). 

Note  5.  Many  practical  problems,  such  as  finding  areas,  lengths  of  curves, 
volumes  and  surfaces  of  solids,  and  so  on,  can  be  reduced  to  finding  the  limit 
of  the  sum  of  an  infinite  number  of  infinitesimals  of  the  form  f(x)  dx.  (See 
Arts.  Ill,  112,  135-140.)  As  has  been  seen  above,  the  anti-differential 
of  /(x)  dx  is  of  great  service  in  determining  this  limit ;  accordingly,  con- 
siderable attention  must  be  given  to  mastering  methods  for  finding  anti- 
differentials. 

Note  6.  The  process  of  finding  the  anti-differential  of  f(x)  dx  is  nearly 
always  more  difficult  than  the  direct  process  of  differentiation,  and  frequently 
the  deduction  of  an  anti-differential  is  impossible.  Wlien  the  anti-differential 
of  /(x)  dx  cannot  be  found  in  a  finite  form  in  terms  of  ordinary  functions, 
approximate  values  of  the  definite  integral  can  be  found  by  methods  dis- 
cussed in  Chapter  XIV.  The  impossibility  of  evaluating  the  first  member  of 
(8)  in  terms  of  the  ordinary  functions  has  sometimes  furnished  an  occasion 
for  defining  a  new  function,  whose  properties  are  investigated  in  higlier 
mathematics.     (On  this  point  see  Snyder  and  Hutchinson,  Calculus,  Art.  123, 


158 


INFINITESIMAL   CALCUL US. 


[Ch.  X. 


foot-note.)  For  instance,  the  subject  of  Elliptic  Functions  arose  out  of  the 
study  of  what  are  called  the  elliptic  integrals  (see  Art.  137,  Ex.  4,  Art.  174, 
Note  4,  Art.  122,  Note  4). 

(The  ordinary  elementary  functions  can  be  defined  by  means  of  the 
calculus,  and  their  properties  thence  developed.) 

Note  7.  At  the  beginning  of  this  article  the  principle  was  enunciated 
that  the  area  bounded  by  a  smooth  curve  PQ  (Fig.  41),  the  ic-axis,  and  a  pair 
of  ordinates,  is  the  limit  of  the  sum  of  certain  inner,  or  outer,  rectangles 
constructed  between  the  ordinates.  The  student  can  easily  show  that  this 
principle  holds  for  the  smooth  curves  in  Figs.  42  a,  b,  c. 


A  0  B  X 

Fig.  42  a. 


B   X 


N 
Fig.  42  c. 


Note  8.  This  article  shows  that  a  definite  integral  may  be  represented 
geometrically  as  an  area.  For  a  general  analytical  exposition  of  integration 
as  a  summation,  see  Snyder  and  Hutchinson,  Calculus,  Art.  148.  Their 
exposition  depends  on  Taylor's  theorem  (Art.  176).  Also  see  the  references 
mentioned  in  Art.  97,  Note  5. 

Ex.  Show  that  the  calculus  method  of  computing  the  area  in  Fig.  42  c 
bounded  by  PMNRQ,  AB,  AP,  and  BQ  really  gives  area^PJf  +  area  J?  ^B 
—  area  MNR. 

[As  a  point  moves  along  the  curve  from  P  to  Q,  dx  is  always  positive.  In 
APM  y  is  positive,  in  MNB  negative,  in  BQB  positive.  Accordingly,  the 
elements  of  area,  /(x)  dx  or  y  dx,  are  positive  in  APMdi.nd.RQBy  and  negative 
in  MNB.I 

EXAMPLES. 


N.B.  The  knowledge  already  obtained  in  Chapter  IV.  about  anti-differen- 
tials is  sufficient  for  the  solution  of  the  following  examples.  It  is  advisable 
to  make  drawings  of  the  curves  and  the  figures  whose  areas  are  required. 


1.    Find  the  area  between  the  cubical  parabola  y 
X-axis,  and  the  ordinates  for  which  a;  =  1,  x  =  3. 


x8  (Fig.,  p. 412),  the 


96.]  INTEGRATION.  159 

According  to  (3)  and  (8),  the  area  required  =  i  x^dx 

=  20  sq.  units  of  area. 

2.  Find  the  area  between  the  curve  in  Ex.  1,  the  x-axis,  and  the  ordi- 
nates  for  which  x  =  —  2,  x  =  S.  Ans.  16i  sq.  units. 

3.  Explain  the  apparent  contradiction  between  the  results  in  Exs.  1,  2. 

4.  Find  the  actual  number  of  square  units  in  the  figure  whose  boundaries 
are  given  in  Ex.  2.  Ans.  24i  sq.  units. 

5.  Find  the  area  between  the  parabola  2y  =  7  x^,  the  ic-axis,  and  the 
ordinates  for  which  :   (1)  x  =  2,  a;  =  4  ;  (2)  a:  =  —  3,  x  =  5. 

Ans.    (1)  651  sq.  units  ;  (2)  177^  sq.  units. 
N.B.   A  table  of  square  roots  will  save  time  and  trouble. 

6.  Find  the  area  between  the  parabola  y^  =  Sx,  the  cc-axis,  and  the 
ordinates  for  which  :  (1)  x  =  0,  x  =  S  ;  (2)  x  =  2,  x  =  7. 

Ans.    (1)  9.798  sq.  units  ;  (2)  29.59  sq.  units. 

7.  Find  the  area  of  the  figure  bounded  by  the  parabola  y'^  =  6x  and 
the  chord  perpendicular  to  the  x-axis  at  x  =  4.  Ans.  26.128  sq.  units. 

8.  Find,  by  the  calculus,  the  area  bounded  by  the  line  y  =  8  x,  the 
X-axis,  and  the  ordinate  for  which  x  =  4.  Ans.    24  sq.  units. 

9.  (1)  Find,  by  the  calculus,  the  area  of  the  figure  bounded  by  the  line 
y  =  Sx,  the  x-axis,  and  the  ordinates  for  which  x  =  4,  x  =  —  4.  (2)  How 
many  sq.  units  of  gold  leaf  are  required  to  cover  this  figure  ? 

Ans.    (1)  0  ;  (2)  48  sq.  units. 

10.  (1)  Find  the  area  between  a  semi-undulation  of  the  curve  y  =  sin  x 
and  the  x-axis.  (2)  Find  the  area  of  the  figure  bounded  by  a  complete 
undulation  of  this  curve  and  the  x-axis.  (3)  How  many  sq.  units  of  gold- 
leaf  are  required  to  cover  this  figure.  Ans.    (1)  2  ;  (2)  0  ;  (3)  4. 

11.  Compute  the  area  enclosed  by  the  parabola  y^  =  4  x  and  the  lines 
X  =  2,  X  =  5.  Ans.    22.27  sq.  units. 

12.  Compute  the  area  enclosed  by  the  parabola  y  =  x'^  and  the  lines 
y  =  1,  y  =  4.  Ans.    9^  sq.  units. 

13.  Find  the  area  between  the  parabolas  x^  =  y  and  y^  =  Sx. 

Ahs.    2|  sq.  units. 

14.  Find  the  area  between  the  curves  :  (1)  y'^  —  x  and  y^  —  x^\  (2)  x^  =  y 
and  ?/2  —  x^.     (Make  figures.)  Ans.    (1)  y*j  sq.  units  ;  (2)  J-  sq.  units. 

15.  Find  the  area  bounded  by  the  curves  in  Ex.  14  (2)  and  the  lines 
X  =  2,  X  =  4.  Ans.    8.129  sq.  units. 

]V.B.   Art.  Ill  may  be  taken  up  now. 


160  INFINITESIMAL    CALCULUS.  [Ch.  X. 

97.  Integration  as  the  inverse  of  differentiation.  The  indefinite 
integral.  Constant  of  integration.  Particular  integrals.  In  many 
cases  there  is  required,  not  the  limit  of  the  sum  of  an  infinite 
number  of  infinitesimals  of  the  form  f(x)dx,  but  the  function 
whose  derivative  or  differential  is  given.  The  following  is  an 
instance  from  geometry.  When  a  curve's  equation,  y=f(x),  is 
known,  differentiation  gives  the  slope  at  any  point  on  the  curve 

in  terms  of  the  abscissa  x,  namely,  -^=f'(x)  (Art.  24).     On  the 

other  hand,  if  this  slope  is  given,  integration  affords  a  means  of 
finding  the  equation  of  the  curve  (or  curves)  satisfying  the  given 
condition  as  to  slope.  Again,  an  instance  from  mechanics :  if  a 
quantity  changes  with  time  in  an  assigned  way,  differentiation 
determines  the  rate  of  change  for  any  instant  (Art.  25).  On  the 
other  hand,  if  this  rate  of  change  is  known,  integration  provides 
a  means  for  determining  the  quantity  in  terms  of  the  time.  (See 
Art.  22,  Notes  1,  2,  and  Art.  27  a.) 


EXAMPLES. 

Ex.  1.   The  slope  at  any  point  (x,  y)  of  the  cubical  parabola  y  =  x^  is  Sx^  ; 

that  is,  at  all  points  on  this  curve,  -^  =  Sx^  and  dy  =  3x^  dx. 

dx 

Now  suppose  it  is  known  that  a  curve  satisfies  the  following  condition^ 

namely,  that  its  slope  at  any  point  (oj,  ?/)  is  3  a;^ ;  i.e.  that  for  this  curve, 

^  =  3  x2,  (whence,  dy=^Zx'^ dx). 
dx 

Then,  evidently,  y  =  x^  +  c, 

in  which  c  is  a  constant  which  can  take  any  arbitrarily  assigned  value.  This 
number  c  is  called  a  constant  of  integration;  its  geometrical  meaning  is 
explained  in  Art.  99.  Since  c  denotes  any  constant,  there  is  evidently  an 
infinite  number  of  curves  (cubical  parabolas,  ?/  =  x^  +  2,  y  =  x^  —  10,  ?/  =  x^ 
+  7,  etc.,  etc.)  which  satisfy  the  given  condition.  If  a  second  condition  is 
imposed,  the  constant  c  will  have  a  definite  and  particular  value.  For 
instance,  let  the  curve  be  required  to  pass  through  the  point  (2,  1).  Then, 
1  =  23  -f-  c ;  whence  c  =  —  7,  and  the  equation  of  the  curve  satisfying  both 
the  conditions  above  is  y  =  x^  —  1.     (Also  see  Ex.  17,  Art.  37.) 

2.    Suppose  that  a  body  is  moving  in  a  straight  line  in  such  a  way  that 
(the  number  of  units  in)  its  distance  from  a  fixed  point  on  the  line  is  always 


97.]  INTEGRATION.  161 

(the  number  of  units  in)  the  logarithm  of  the  number  of  seconds,  t  say,  since 

the  motion  began  ;  i.e.  so  that 

s  =  log  t. 

Then,  the  speed,  ^  =  1    and  ^  =  ^  • 

Now  suppose  it  is  known  that  at  any  time  after  the  beginning  of  its 
motion,  after  t  seconds  say,  the  speed  of  a  moving  body  is  -;  i.e.  that 


=  -,  [whence,  ds=-) 


Then,  evidently,  s  =  log  «  +  c, 

in  which  c  is  an  arbitrary  constant.  If  a  second  condition  is  imposed,  the 
constant  c  will  take  a  definite  value.  For  instance,  let  the  body  be  4  units 
from  the  starting-point  at  the  end  of  2  seconds,  i.e.  let  s  =  4  when  t  =  2. 

'^^^^  4  =  log  2  +  c  ;  whence  c  =  4  -  log  2, 

and  s  =  log  ^  +  4  —  log  2. 

3.  In  Ex.  1  determine  c  so  that  the  cubical  parabola  shall  go  through 
(a)  the  point  (0,  0);  (6)  the  point  (7,  -4);  (c)  the  point  (-8,  2);  {d)  the 
point  (A,  k).     Draw  the  curves  for  (a),  (6),  (c). 

4.  Find  the  curves  for  which  the  slope  at  any  point  is  4.  Determine 
the  particular  curves  which  pass  through  the  points  (0,  0),  (2,  3),  (—7,  1), 
respectively.     Draw  these  curves. 

5.  Find  the  curves  for  which  (the  number  of  units  in)  the  slope  at 
any  point  is  8  times  (the  number  of  units  in)  the  abscissa  of  the  point. 
Determine  the  particular  curves  which  pass  through  the  points  (0,  0),  (1,  2), 
(2,  3),  (  —  4,  2),  respectively.     Draw  these  curves. 

6.  How  are  the  curves  in  Exs.  4,  1,  3,  5,  respectively,  affected  when 
the  constants  of  integration  are  changed? 

7.  If  at  any  moment  the  velocity  in  feet  per  second  at  which  a  body 
is  falling  is  32  times  the  number  of  seconds  elapsed  since  it  began  to  fall  from 
rest,  what  is  the  general  formula  for  its  distance,  at  any  instant,  from  a  point 
on  the  line  of  fall  ? 

In  this  instance,       —  =  32  «,  (whence,  ds  =  d!it  dt). 
dt 

Hence  s  =  iet^  +  c. 

8.  In  Ex.  7,  at  the  end  of  t  seconds  what  is  the  distance  measured 
from  the  starting-point  ?  What  is  the  distance  at  the  end  of  2  seconds  ?  of 
4  seconds  ?  of  5  seconds  ?  What  are  the  distances,  in  these  respective  dis- 
tances, measured  from  a  point  10  feet  above  the  starting-point  ?  If  at  the 
time  of  the  beginning  of  fall,  the  body  is  20  feet  below  the  point  from  which 


162  INFINITESIMAL   CALCULUS.  [Ch.  X. 

distance  is  measured,  what  is  its  distance  below  this  point  at  the  end  of  t 
seconds  ?  Explain  the  meaning  of  the  constant  of  integration  in  the  general 
formula  derived  in  Ex.  7  ?  Derive  the  results  in  Ex.  8  from  this  general 
formula. 

Suppose  that  cl<f>(x)=f(x)dXy  (1) 

then  also  (Art.  29),    d \  cf>(x)  +  c |  =  f(x)dx,  (2) 

in  which  c  is  any  constant.  Hence,  if  <f>(;x)  is  an  anti-differential 
of  f(x)dx,  <t>(x)  H-  c  is  also  an  anti-differential  of  f{x)dx.     That  is, 

if  d</)(x)  =f(x)dx, 

then  ^f(Qo)tlac  =  ^(a^)  +  c,  (3) 

in  which  c  is  an  arbitrary  constant.  Thus  the  anti-differential  of 
f(x)dx  is  indefinite,  so  far  as  an  added  arbitrary  constant  is  con- 
cerned. (This  has  already  been  pointed  out  in  Art.  29,  Note  6.) 
On  this  account  the  anti-differential  is  called  the  indefinite  integrral. 
The  arbitrary  constant  is  called  the  constant  of  integration.  The 
indefinite  integral  is  often  called  the  general  integral.  If  the 
constant  of  integration  be  given  a  particular  value,  as  ^,  —2, 
100,  etc.,  the  integral  is  called  a  particular  integral.    For  instance, 

the  indefinite,  or  general,  integral  of  x''dx,  i.e.    |  x^dx  is  \x^-\-c', 

and  particular  integrals  of  x^dx  are  i  cc^  +  5,  \x^  — 11,  etc. 

9.    Name  the  indefinite  (or  general)  integrals  and  the  particular  integrals 
appearing  in  Exs,  1-8. 

10.    How  many  particular   integrals    (anti-differentials)    can  a  function 
have  ?     What  must  the  difference  between  any  pair  of  them  be  ? 

Note  1.     It  should  be  noted  that  the  indefiniteness  in  the  integral  does 
not  extend  to  the  terms  involving  the  variable.     For  instance. 


J' 


and     f  (X  4  \)dx  =  ( (x  +  \)d(x  -f  1)*  =  \{x  -\- \y -{■  k  =  lofi -{- x -{- I -\- k  \ 
thus  the  terms  involving  x  are  the  same. 

Note  2.     The  origin  of  the  words  integral  and  integration  has  been 
indicated  in  Art.   94.    It  is,  in  a  measure,  to  be  regretted  that  the  term 

integral  and  the  symbol    i  ,  which  both  imply  summation,  should  also  be 
used  to  denote  an  anti-differential.     In  accordance  with  the  fashion  in  vogue 

*  Since  d(x  +  1)  =  c?x. 


98.]  INTEGRATION.  163 

in  trigonometry  for  denoting  inverse  functions  {e.g.  since  and  sin-i  x  for  sine 
of  X  and  anti-sine,  or  inverse  sine,  of  aj,  respectively*)  the  anti-derivative 
of /(x)  and  the  anti-differential  of  f(x)dx  are  sometimes  denoted  by  D-^f{x) 

and  d-^f{x)dx  respectively.  Thus  \f{x)dx,  d-Y(x)dx,  and  D~Y(x),  are 
equivalent. 

Note  3.     K  d(p(x)  =f(x)dx,  then  (Art.  96)    Cf(x)dx  =  0(a;)  -  0(a). 

If  the  upper  end-value  x  is  variable,  and  the  lower  end-value  a  is  arbitrary, 
then  this  integral  is  indefinite  and  of  the  form  0(x)  -f  c.  Accordingly,  the 
indefinite  integral  may  be  regarded  as  in  the  form  of  a  definite  integral  whose 
upper  end-value  is  the  variable,  and  whose  lower  end-value  is  arbitrary. 

Note  4.  Result  (8),  Art.  96  for  the  area  of  APQB  (Fig.  41)  can  also  be 
derived  by  a  method  which  is  founded  on  the  notion  of  the  indefinite  integral. 
For  instance,  see  Todhunter,  Integral  Caladus,  Art.  128,  or  Murray,  Integral 
Calculus,  Art.  13. 

Note  5.  References  for  collateral  reading  on  the  notions  of  integra- 
tion, definite  integral,  and  indefinite  integral.  Gibson,  Calculus,  §§  82,  110, 
124-126  ;  Williamson,  Integral  Calculus,  Arts.  1,  90,  91,  126  ;  Harnack, 
Calculus  (Cathcart's  translation),  §§  100-106  ;  Echols,  Calculus,  Chap.  XVI. ; 
Lamb,  Calculus,  Arts.  71,  72,  86-93. 

98.  Geometric  or  graphical  representation  of  definite  integrals. 
Properties  of  definite  integrals.  It  has  been  seen  (Art.  96)  that  if 
PQ,  (Fig.  41)  is  the  curve  whose  equation  is 


then  the  integral  j   f{x) 


dx 


gives  the  area  bounded  by  the  curve,  the  cc-axis,  and  the  ordinates 
for  which  x  =  a  and  x  =  b  respectively.  Accordingly,  the  figure 
thus  bounded  may  be  said,  and  may  be  used,  to  represent  the 
integral  graphically.     Hence,  in  order  to  represent  an  integral, 

I     <^  (x)  dx  say  (no  matter  whether  this  integral  be  an  area,  or  a 

length,  or  a  volume,  or  a  mass,  etc.),  draw  the  curve  whose 
equation  is  y  =  <f>  (x),  and  draw  the  ordinates  for  which  x  —  l  and 
x  =  m  respectively.  The  figure  bounded  by  the  curve,  the  ic-axis, 
and  these  ordinates,  is  the  graphical  representative  of  the  integral, 
and  (Art.  96)  the  number  of  units  in  the  area  of  this  figure  is  the 
same  as  the  number  of  units  in  the  integral. 

*  See  Art.  12,  Note, 


164 


INFINITESIMAL   CALCUL US, 


[Ch.  X. 


TTie  foUoicing  properties  of  definite  integrals  are  important.  Prop- 
erties (h)  and  (c)  are  eg^^ily  deduced  by  using  the  graphical 
representatives  of  the  integrals. 

{a)    If  dct>(x)=f(x)dx,  then  (Art.  96) 

f{x) dx  =  (f>{b)  —  cfi (a)     and      I  f(x) dx  =  <f>{a)  —  <f){b)', 

f(x)dx  =  —  I   f{x)dx. 

Therefore,  if  the  end-values  of  the  variable  in  an  integral  be 
interchanged,  the  algebraic  sign  of  the  integral  will  be  changed. 

Ex.   Give  several  concrete  illustrations  of  this  property. 

f(x)dx=  I   f{x)dx-\-  I   f(x)dx,  whatever  c  may  be. 

Draw  the  curve  y=f(x),  and  draw  ordinates  AP,  BQ,  CR,  for 
which  x  =  ay  x  =  b,  x  =  c,  respectively.     Then : 


Fig.  43  a. 
In  Fig.  43  a, 


Fig.  43  6. 
In  Fig.  43  6, 


Cfix) dx  =  area  APQB  C  f(x) dx  =  area  APQB 

=  area  APBG  -f  area  CR^B  =  area  APRC  -  area  BQ,RQ 

=  rf(x)dx-h  rf(x)dx, 

•/a  •/c 


98.]  INTEGRATION,  166 

Similarly,  it  can  be  shown  that 

f /(a;)  dx  =  f /(^)  dx  +  jT /(^')  cZaj  +  •  •  •  +  J^ /(»)  <^^  +  jy^^^  ^^* 

That  is,  a  definite  integral  can  be  broken  up  into  any  number  of 
similar  definite  integrals  that  differ  only  in  their  end-values. 
(Similar  definite  integrals  are  those  in  which  the  same  integrand 
appears.) 

Ex.  1.    Prove  the  principle  just  enunciated. 

Ex.  2.    Give  concrete  illustrations  of  the  principles  in  (6). 

(c)    The  mean  value  of  f(x)  for  all  values  of  x  from  a  to  b. 
(That  is,  the  mean  value  of  f{x)  when  x  varies  continuously 


from  a  to  6.)     Draw  the  curve  y=f(x),  and  at  A  and  B  erect 
the  ordinates  for  w^hich  x  =  a  and  x  =  b  respectively.     Then 


Cf(x)dx  =  area  APQB. 


Now,  evidently,  on  the  base  AB  there  can  be  a  rectangle  whose 
area  is  the  same  as  the  area  of  APQB,  Let  ALMB,  which  has 
an  altitude  CB,  be  this  rectangle ;  then 

I  f(x)  dx  =  area  ALMB  =  area  AB  •  CR 

^  (6 -a),  length  Ci?.  (1) 


166  INFINITESIMAL   CALCULUS.  [Ch.  X. 

The  length  CR  is  said  to  be  the  mean  value  of  the  ordinates 
f{x)  from  X  =  a  to  X  =  h.     Hence,  from  (1), 


Mean  value  of  /(a?)  from  ^  ^  j^/W<?^ 
nc  —  a  id  oc  =  b  ]  ~      h  —  a 


(2) 


In  words,  the  mean  value  off(x)  when  x  varies  continuously  from 
a  to  b,  is  equal  to  the  integral  of  f{x)  dx  from  the  end-value  a  to 
the  end-value  b,  divided  by  the  difference  between  these  end-values. 

EXAMPLES. 

1.  Make  a  graphical  representation  of  each  of  the  integrals  appearing 
in  Exs.  2-5  below. 

2.  Find  the  mean  length  of  the  ordinates  of  the  parabola  y  =  x^  from 

a;  =  1  to  a:  =  3.  ^3 

\  x^dx 

Mean  length  =  ^ =  4i 

^  3-1  ' 

3.  In  the  parabola  y  —  x^,  find  the  mean  length  of  the  ordinates  of  the 
arc  between  a;  =  0  and  a:  =  2  ;  and  find  the  mean  length  of  the  ordinates 
from  X  =  —  2  to  x  =  2.  Explain,  with  the  help  of  a  figure,  why  these  mean 
lengths  are  the  same. 

4.  In  the  cubical  parabola  y  =  x^. 

5.  In  the  line  y  =  ix. 

99.  Geometric  (or  graphical)  representation  of  indefinite  integrals. 
Geometric  meaning  of  the  constant  of  integration.     If 

d<f>(x)  =f(x)  dx, 

then  (Art.  97)  Cf(x)  dx  =  <ji{x)  +  c,  (1) 

in  which  c  is  an  arbitrary  constant.     Draw  the  curve 

y  =  <l>{x)',  (2) 

let  AB  be  the  curve.     Give  c  the  particular  values  2  and  10,  and 
draw  the  curves,  y  =  <^{x)  +  2  (3) 

and  y  =  <f>(x)  -f- 10.  (4) 

*For  clear  proof  that  this  is  the  mean  value,  see  Art.  141,  where  the 
topic  of  mean  values  is  more  fully  discussed,  and  Echols,  Calcidus,  Art.  150 
(and  Arts.  151,  152). 


99.] 


INTEGRATION. 


167 


Let  CD  and  EF  be  these  curves.     In  the  case  of  each  one  of  the 
curves    obtained    by   giving 
particular  values  to  c, 


dx 


f{^); 


Fig.  45. 


and  hence,  at  points  having 
the  same  abscissa  the  tan- 
gents to  these  curves  have 
the  same  slope,  and,  accord- 
ingly, are  parallel.  For  in- 
stance, on  each  curve,  at 
the  point  whose  abscissa  is 
m  the  slope  of  the  tangent  is  f{m). 

Moreover,  the  distance  between  any  two  curves  obtained  by 
giving  c  particular  values,  measured  along  any  ordinate,  is  always 
the  same.  For,  draw  the  ordinates  KR  and  ST  at  x  =  m  and 
X  =  n,  respectively,  as  in  the  figure.     Then,  by  Equations   (3) 

and  (4),         MK=  </)(m)  +  2 ;  NS  =  <f>(n)  +  2 ; 

and 


ME  =  ct>{m)  +  10 ;  NT=  ^(n)  -\- 10. 


Hence 


KE  =  8, 


and    ST=S. 


Accordingly,  the  graphical  representation  of  the  indefinite  integral, 

I  f(x)  clx,  consists  of  the  family  of  curves,  infinite  in  number, 

whose  equations  are  of  the  form  y  =  <f>(x)  +  c,  and  which  are 
severally  obtained  by  giving  c  particular  values ;  and  the  effect  of 
changing  c  is  to  move  the  curve  in  a  direction  parallel  to  the 
?/-axis.     (Also  see  Art.  29,  Note  2.) 

Ex.  1.  How  many  different  values  can  be  assigned  to  c  ?  How  many 
particular  integrals  are  included  in  the  general  integral  ?  How  many  different 
curves  can  represent  the  indefinite  integral  ? 

Ex.  2.    Write  the  equations  of  several  curves  representing  each  of  the 

following  integrals,  viz.:  ixdx,  ix'^dx,  \Sxdx,  \Sdx,  f(2x  +  5)dx. 
Draw  the  curves. 


168 


INFINITESIMAL   CALCULUS, 


[Ch.  X. 


(1) 


100.   Integral  curves.     If  d  <\>{x)  =f(x)  dx, 

then  (Art.  96)  f /(a^)  d^  =  <{^(^)  -  <f>(0). 

The  curve  whose  equation  is 

y  =  4>(x)  -  <t>{0),  i.e.  y=  \    /(«)  dx, 

which  is  one  of  the  particular  curves  representing  y  =  <ji{x)  4-  c 
(see  Art.  99),  is  called  the  first  integral  curve  for  the  curve  y  =f(x). 
Since  the  area  of  the  figure  bounded  by  the  curve  y=f(x),  the 
flj-axis,  and  the  ordinates  at  a?  =  0  and  x  =  x,  is  <f}(x)  —  <f>(0)  (Art. 
96),  the  number  of  units  of  length  in  the  ordinate  at  the  point  of 
abscissa  x  on  the  curve  (1),  is  the  same  as  the  number  of  units 
of  area  in  this  figure.  Accordingly,  if  the  first  integral  curve  of 
a  given  curve  be  drawn,  the  area  bounded  by  the  given  curve,  the 
axes,  and  the  ordinate  at  any  point  on  the  avaxis,  can  be  obtained 
merely  by  measuring  the  length  of  the  ordinate  drawn  from  the 
same  point  to  the  integral  curve.  Consequently,  it  may  be  said 
that  this  ordinate  graphically  represents  the  area,  and  thus,  the 
integral.  ^ 

Note  1 .     The  original  curve  y  =  f(x)  is  the  derived  or  differential  curve 
of  curve  (1). 

Ex.    For  instance,  for  the  line      y  =  ^x  +  S;  (2) 

since  P  (^  x  +  3)  dx  =  |-  x^  -f  3  a;, 

the  first  integral  curve  of  curve  (2)  is  the  parabola  y  =  lx^  -}-Sx.  (3) 

These  two  curves  are  shown 
here.  If  M  be  any  point  on  the 
X-axis,  and  OM=m  units  of  length, 
and  the  ordinate  3ILG  be  drawn, 

(the  number  of  units  of  length 
in  MG)  =  (the  number  of  units  of 
area  in  OKLM). 

For,  length  MG,  by   (3),  is  J  m^ 
+  3m;  and 

area  OKLM 


100,  101.]  INTEGRATION.  169 

Just  as  a  given  curve  —  it  may  be  called  the  original  or  the 
fundamental  curve  —  has  a  first  integral  curve,  this  first  integral 
curve  also  has  an  integral  curve.  The  latter  curve  is  called  the 
second  integral  curve  of  the  fundamental  curve.  Again,  the  second 
integral  curve  has  an  integral  curve ;  this  is  said  to  be  the  third 
integral  curve  of  the  fundamental  curve.  On  proceeding  in  this 
way  a  system  of  any  number  of  successive  integral  curves  may 
be  constructed  belonging  to  a  given  fundamental  curve. 

Note  2.  The  integral  curve  can  be  drawn  mechanically  from  its  funda- 
mental by  means  of  an  instrument  called  the  integraph,  invented  by  a 
Russian  engineer,  Abdank-Abakanowicz. 

Note  3.  Integral  curves  are  of  great  assistance  in  obtaining  graphical 
solutions  of  practical  problems  in  mechanics  and  physics.  For  further  in- 
formation about  integral  curves  and  their  uses  and  the  theory  of  the  integraph, 
and  for  other  references,  see  Gibson,  Calculus,  §§  83,  84  ;  Murray,  Integral 
Calculus,  Art.  15,  Chap.  XII.,  pp.  190-200  (integral  curves),  Appendix, 
Note  G  (on  integral  curves),  pp.  240-245  ;  M.  Abdank-Abakanowicz,  Les 
Integraphes :  la  courhe  integrate  et  ses  applications  (Paris,  Gauthier-Villars), 
or  BitterlVs  German  translation  of  the  same,  with  additional  notes  (Leipzig, 
Teubner).  Also  see  catalogues  of  dealers  in  mathematical  and  drawing 
instruments. 

EXAMPLES. 

1.  Show  that,  for  the  same  abscissa,  the  number  of  units  of  length  in 
the  ordinate  of  the  fundamental  curve  is  the  same  as  the  number  of  units  in 
the  slope  of  its  first  integral  curve. 

2.  Does  the  first  integral  curve  belong  to  the  family  of  curves  referred  to 
in  Art.  99  ? 

3.  Show  how  the  members  of  the  family  of  curves  in  Art.  99  may  be 
easily  drawn  when  an  integraph  is  available. 

4.  Write  the  equations  of  the  first,  second,  and  third  integral  curves 
of  the  following  curves  :  (a)  y  =  x  ;  (b)  y  =  2x  +  b;  (c)  y  =  smx;  (d)  y  =  e*. 
Draw  all  these  fundamental  and  integral  curves.  Can  the  curve  x^y  =  1  be 
treated  in  a  similar  manner  ? 

5.  Find  and  draw  the  curve  of  slopes  for  each  of  the  curves  (a),  (6), 
(c),  (d),  Ex.  4.  Then  find  and  draw  the  first,  second,  and  third  integral 
curves  of  each  of  these  curves  of  slope. 

101.  Summary.  The  two  processes  of  the  infinitesimal  calculus, 
namely,  differentiation  and  integration,  have  now  been  briefly 
described. 


170  INFINITESIMAL   CALCULUS.  [Ch.  X. 

The  process  of  differentiation  is  used  in  solving  this  problem, 
among  others :  the  function  of  a  variable  being  given,  find  the 
limiting  value  of  the  ratio  of  the  increment  of  the  function  to  the 
increment  of  the  variable  when  the  increment  of  the  variable 
approaches  zero  (Art.  22).  This  problem  is  equivalent  to  finding 
the  ratio  of  the  rate  of  increase  of  the  function  to  the  rate  of 
increase  of  the  variable  (Art.  2Q).  If  the  function  be  represented 
by  a  curve,  the  problem  is  equivalent  to  finding  the  slope  of  the 
curve  at  any  point  (Art.  24). 

The  process  of  integration  may  be  regarded  as  either : 

(a)  a  process  of  summation  ;  or 

(b)  a  process  which  is  the  inverse  of  differentiation. 

Integration  is  used  in  solving  both  of  the  following  problems, 
viz. : 

(1)  To  find  the  limit  of  the  sum  of  infinitesimals  of  the  form 
f(x)  dx,  X  being  given  definite  values  at  which  the  summation 
begins  and  ends  (Arts.  94-96) ; 

(2)  To  find  the  anti-differential  of  a  given  differential  f{x)  dx 
(Art.  97). 

Problem  (1)  is  equivalent  to  finding  a  certain  area;  problem 
(2)  is  equivalent  to  finding  a  curve  when  its  slope  at  every  point 
is  known. 

In  solving  problem  (1)  the  anti-differential  oif{x)  dx  is  required 
(Art.  96).  Hence,  in  both  problems  (1)  and  (2)  it  is  necessary  to 
find  the  anti-differentials  of  various  functions  of  the  form/(ic)  dx. 
Chapters  XI.  and  XIII.  are  devoted  to  showing  how  anti-differ- 
entials may  be  found  in  the  case  of  several  of  the  comparatively 
small  number  of  functions  for  which  this  is  possible.  It  may  be 
stated  here  that,  in  general,  integration  is  more  difficult  than  the 
direct  process  of  differentiation. 


CHAPTER  XI. 

ELEMENTARY   INTEGRALS. 

102.  In  this  chapter  the  elementary  or  fundamental  integrals 
(anti-differentials)  are  obtained,  and  some  general  theorems  and 
particular  methods  which  are  useful  in  the  process  of  anti-differ- 
entiation are  described.  There  is  one  general  fundamental  process 
(Art.  22)  by  which  the  differential  of  a  function  can  be  obtained. 
On  the  other  hand,  there  is  no  general  process  by  which  the  anti- 
differential  of  a  function  can  be  found.*  The  simplest  integrals, 
which  are  given  in  Art.  103,  are  discovered  by  means  of  results 
made  known  in  differentiation. 

In  Art.  104  certain  general  theorems  in  integration  are  deduced. 
Two  particular  processes,  or  methods,  of  integration  which  are 
very  serviceable  and  frequently  used,  are  described  in  Arts.  105, 

106.  A  further  set  of  fundamental  integrals  is  derived  in  Art. 

107.  When  f{x)  is  a  rational  fraction  in  x,  the  anti-differential 
of  f(x)dx  may  be  found  by  means  of  the  results  in  Arts.  103, 107; 
for  this  reason  examples  involving  rational  fractions  are  given  in 
Art.  108.  The  integration  of  a  total  differential  is  considered  in 
Art.  109. 

So  far  as  finding  anti-differentials  is  concerned,  this  is  the  most 
important  chapter  in  the  book.  The  student  is  strongly  recom- 
mended to  make  himself  thoroughly  familiar  with  the  chapter 
and  to  work  a  large  number  of  examples,  so  that  he  can  apply  its 
results  readily  and  accurately.  T7ie  list  of  formulas,  I.  to  XX  VL 
(Arts.  103,  107),  should  be  memorized.  Every  function,  f(x)dx, 
whose  integral  can  be  expressed  in  finite  form  in  terms  of  the 
functions  in  elementary  mathematics,  is  reducible  to  one  or  more 
of  the  forms  in  this  list.  It  is  often  necessary  to  make  reductions 
of  this  kind.     A  ready  knowledge  of  these  forms  is  not  only  useful 

*  There  is  a  general  process  by  which  the  value  of  a  definite  integral  can 
be  found  approximately,  as  described  in  Art.  123. 

171 


172  INFINITESIMAL   CALCULUS.  [Ch.  XI. 

for  integrating  them  immediately  when  presented,  but  is  also  a 
great  aid  in  indicating  the  form  at  which  to  aim,  when  it  is  neces- 
sary to  reduce  a  complicated  expression. 

103.  Elementary  integrals.  The  following  formulas  in  integra- 
tion come  directly  from  the  results  in  Arts.  37-55,  and  can  be 
verified  by  differentiation.  Here  w  denotes  a  function  of  any 
variable,  and  c,  Co,  c^  denote  arbitrary  constants. 

I.    (u^^du  =  — h  c,    in  which  7i  is  a  constant. 

J  n  +  1        ' 

Note  1.  This  result  is  applicable  in  the  case  of  all  constant  values  of  w, 
excepting  n  =—  I.     The  latter  case  is  given  in  II. 

II.    f ^  =  logu  +  co  =  log «*  +  log c  =  log cu. 
J  u 

Note  2.  The  various  ways  in  which  the  constant  of  integration  can 
appear  in  this  integral,  should  be  noted. 

NoTK  3.  Formula  II.  can  also  be  derived  by  means  of  I.  (See  Murray, 
Integral  Calculus,  p.  37,  foot-note.) 

'  ni.    (a^du  =  -^—  +  e. 

J  log  a 

IV.  ie^'du  =  e^'  +  c. 

V.  (  sin  u  du  =  -  cos  «*  +  c. 

VI.  (  cos udu  =  ^mu  +  c, 

VII.  (  sec^  udu  =  tan  u  -\-  c, 

VIII.  j  csc^ u  du  =  -  coin  -\-  c, 

IX.  isecutSLnudu  =  secu-i-€, 

X.  \  CSC  u  cot  udu=  — CSC  u-\- c. 

XI.  f_^_=sm-i«*  +  c  =  -cos-i«*-i-Ci. 

[Remark.  By  trigonometry  sin-i  w  =  —  cos-i  m  +  2  wtt  -}-  -•  See  Art.  97, 
Ex.  10  and  Note  1.]  ^ 


'    du 

-i&n 

-^u  +  c. 

du 

sec-^M  +  c. 

uVu^- 

1 

du 

vers^M  +  c. 

103,104.]  ELEMENTARY  INTEGRALS.  173 

XU. 

XIII.  f 

XIV.  f 

Note  4,  Integrals  XII.,  XIII.,  XIV.,  may  also  be  written  —  cot'^u  +  c, 
—  csc-i  w  +  c,   —  covers"!  m  +  c,  respectively. 

104.   General  theorems  in  integration. 

A.  Let  f(x),  F(x),  <f>(^x),  •••,  denote  functions  of  x,  finite  in 
number.     By  Arts.  29,  31,  97,  the  differentials  of 

j"[/(ic)  +  1^(«;+  <|>(a?)  +  -.Ociic  +  Co   and 

(fix)dijc  -{-(F(x)dac  +  r<t>(ic)<?a?  +  ■  -  +  ci 

are  each  /(x)  dx  -j-  i^(x)  c^x*  -f-  <^(x)da;  -f-  •  •  •. 

Hence,  the  integral  of  the  sum  of  a  finite  number  of  functions  and 
the  sum  of  the  integrals  of  the  several  functions  are  the  same  in  the 
terms  depending  on  the  variable,  and  can  differ  at  most  only  hy  an 
arbitrary  constant. 

(For  integration  of  the  sum  of  an  infinite  number  of  functions,  see 
Art.  172.) 

EXAMPLES. 

1.  i  (x^  +  cosx  +  e^)c?x  =  j  x^dx  +  |  cosxdx  +  \  e'^dx  +  Cq 

=  J  X*  +  sin  X  +  e^  +  c.  (1) 

Note  1.  Each  integral  in  the  second  member  in  Ex.  1  has  an  arbitrary 
constant  of  integration  ;  but  all  these  constants  can  be  combined  into  one. 

2.  \  (x^  —  sin  X  +  sec2 x)dx  =  |  x^  .+  cos x  +  tan x  +  c. 

B,  The  differentials  of 

\^nu  dx  +  Co  and  m  \u  dx  +  c\ 

are  each  mudx.     Hence, 

a  constant  factor  can  be  moved  from  either  side  of  the  integration 
sign  to  the  other  ivithout  affecting  the  terms  of  the  integral  which 
depend  on  the  variable.    , 


174  INFINITESIMAL   CALCULUS.  [Ch.  XI. 

C.  The  differentials  of 

i  u  dx  +  Co,  "in  f  —  doc  +  Ci,  —  \mu  doo  +  cg, 

are  each  udx.     Hence, 

the  terms  of  the  integral  which  depend  on  the  variable  are  not  affected, 
if  a.  constant  is  introduced  at  the  same  time  as  a  multiplier  on  one 
side  of  the  integration  sign  and  as  a  divisor  on  the  other. 

Note  2.    Theorems  B  and  C  are  useful  in  simpHfying  integrations. 
3.    (1)    (Sxdx  =  s(xdx  =  ^x"^  +  c. 

^  ^  J  x^      J  -4  +  1  3a:=i 

■   4.     I  2  sin  cc  cZx  =  2  I  sin  xdx  =  —  2  cos  x  -\-  c. 

5.  Tsin  2xdx  =  I  j  2  sin  2  a;  fZx  =  |  (  sin  2  x  d(2  x)  :^  —  |  cos  2  aj  +  c. 

Note  3.  A  factor  involving  the  variable  cannot  he  moved,  or  introduced^ 
in  the  manner  described  in  theorems  B  and  C.  Thus,  xx'^dx  =  \x^ -\r  c; 
but  x\xdx  =  lx^  -\-  c.     Also,    \x'^  dx  =  \x^  -{■  c;  but  -  \x^  dx  =  \  x^  +  c. 

„      f  ^  ,         f  sin  u  ^  r  d  (cos  u)  ,       .  .    , 

6.  \  t&n  udu=  \ du  =  -  \  ~ =  -  log  (cos  u)  +  c 

J  J  cos  u  J      cos  u 

=  log  (sec  u)  +  c. 

„      r      ,      ,         r cos  n  -,         f  fZCsin  u)  ,  ,      ,  .      .    . 

7.  i  cot  u  du  =  i  -. du  =  \  ~^. +  c  =  log  (sni  ii)  +  c. 

J  J  smu  J     sin  u 


)  +  c. 


1         lO 

9.    Write  the  anti-derivatives  of  x',   6x'%  2x*^  4x-l^   5x-i4,    — ,  -  , 

;xi  x^^  6^^,  2^^,  -^,   A,,   _^. 
Vx     Vx3     7Vxio 

2  3 

10.  Write  the  anti-differentials  of  v^  dv,  7  vT^  (^^    —  du,  -^  (?s. 

11.  Find   (ax^dx,    (cVF'dt,    (l^/v^dv,    (r\/w^dio. 


105.]  ELEMENTARY  INTEGRALS.  175 

12.  (^K  f-^,  f^^,  r  f^--'^^  dt. 

Jv      Js  +  2     J7-x«     J4«2_3^^n 

13.  (e'lU,    C^e^dx,    (ie^'^xdx,    (^^dx,    (lO'^'dx. 

14.  j  sin  3  a;  cZx,         4  i  cos  7  x  (?x,         9  I  .sec^  5  x  dx,  fsiii  (x  +  «)  rfx, 
fcos(2x+«)<fe,   Tsec2^^  +  '!:\(Zx. 

15.  fsec2xtair2xfZx,        Tsecf  xtanf  xc?x,        C      ^^      ,       f-  ^^^^     , 
^  ^  -^  Vl  -  f-=        -^  Vl  -  x^ 

r       7  (Zx  r  bx^'dx        C      dv  r  tat         C    2dx         C      dt 

r      dx  r     xdx        r      dx  r       dx 

^  X a/9x2  -  1 '    -^  x2 \/x*  -  1 '    -^  A/6x-9x-i'    -^  a/8 X  -  16 x^ 

16.  r(r^-4)2cZ«,     Wat +  x?)3<Zx,    f  e»*  dx,     f  (cos  ax  +  sin  ?ix)  c?x. 

17.  Express  formula  II.  in  words. 

105.  Integration  aided  by  substitution.  Integration  can  often  be 
facilitated  by  the  substitution  of  a  new  variable  for  some  function 
of  the  given  independent  variable;  in  other  words,  by  changing 
the  independent  variable.  Experience  is  the  best  guide  as  to 
what  substitution  is  likely  to  transform  the  given  expression  into 
another  that  is  more  readily  integrable.  The  advantage  of  such 
change  or  substitution  has  been  made  manifest  in  working  some 
of  the  examples  in  Art.  104,  e.g.  Exs.  5,  6,  7,  8,  etc. 

EXAMPLES. 

1.  j  (x  +  a)"  dfx,  in  which  n  is  any  constant,  excepting  —  1. 
Put  x-\-  a  —  z\  then  dx  =  dz.,  and 

f(x  +  a)«ci.=  r.»d.^_^!l^+c  =  (^ii«l:^+c. 

J  J  W-fl  71+1 

This  may  be  integrated  without  explicitly  changing  the  variable.     For,  since 

dx^dix^a),    r(x  +  a)«c?x=  f  (x  +  a)» d{x  +  a)  =  ^^  "^  ^^""^^  +  c. 
J  J  w  +  1 

2.  f(x  +  a)-idx..f-^^=f^^i^  +  ^  =  log(x  +  a)  +  c. 
J  Jx-\-aJx  +  a 


176  INFINITESIMAL    CALCULUS.  [Ch.  XI. 

3.  r    ^^    . 

•^  x  V'4  +  3  a; 
Put  4  +  3  aj  =  ^"^ ;  then  aj  =  1(^2  —  4),  and  dx  —  ^z  dz.    Hence,  on  denoting 
the  integral  by /,  ^     ,  ir/     i  i     \ 


ilog|^+c  =  ilog.^p|^^  +  a 
^  +  2  V4  +  3X  +  2 


4. 

Put  X  =  a  sin  0.     Then  dx  —  a  cos  ^  c?^,  and 


c  =  sin-i?+  c. 


^  >/a2  -  a;2     -^  Va^  -  a^  sin^  ^     *^  « 

This  integral  may  be  found  by  another  substitution.    For,  put  x  =  az',  then 
dx  :^adz,  and    f     J^=  =  f     ,ad^_^  f-^g- 

=  sin-i  z  +  c  =  sin-i  -  +  c. 
a 

5.     fVrt2-x2dx. 

Put  X  =  a  sin  ^.     Then  dx  =  a  cos  ^  d^ ;  and 
f  Va2_x2(?a;=  C  V  a^  -  a^  sin^  ^ .  a  cos  ^  d^ = a^  fcos2^tZ^=-^  f(l+cos20)de 

^«^(^4-«y^)+c  =  «-(^  +  sin^cos^)  +  c 

=  ^fsin-1^  +  ^  J^"^^:^")  +  c  =  i(«^  sin-i^+XA/«2_a;-^)+c. 
2  \         a     a  ^      a^    )  a 

This  important  integral  may  also  be  obtained  in  other  ways ;  see  Ex.  4, 
Art.  118,  and  Ex.  5,  Art.  106. 


C^^JiH —  (Put  u  =  az.)  Ans.  Itan-i  -^*  +  c. 

J  cfi  +  w2  a  a 


6 

+  u 


7.  r ^^  (Put  u  =  az.)  Ans.  -  sec-i  -  +  c. 

8.  f        ^^  (Putw  =  a0.)  ^9is.    vers-i-+c. 
•^  V2  ait  -  m2  « 


Put  V^TT=2;.    Thenx  +  l=^^  dx=2  0(?^,and  f    ^^'^^    =  CifsLDl?^ 

2  ({z^-  l)dz  =  ^  z(z^  -  3)  +  c  =  |(x  -  2)  Vx  +  1  +  c. 


106. J  ELEMENTARY  INTEGRALS.  177 

10.  f^^dx. 

Put  sin  x  =  t.     Then  cos  x  (^  =  (^^,  cos^  xdx  =  cos^  x  •  cos  a;  (Za;  =  ( 1  -  r-^)  dt. 

=  I  «3(4  _  ^2)  _[.  c  ^  3  sin^a;(4  -  sin2a;). 

11.  I  sin^  X  cos  x  dx,    \  tan^  x  sec*  x  dx,    i  sec-  (4  —  7  x)  dx,    f  g-^^  cZx. 

^2-    irXT^'    J-^^-^cZx,     f-^c^.,    fx(x-2)^... 

13.  r  v(x+a)^dx,  rc/o«+/ix)3dx,  f_— ^_-,  f — ^y 

*^  -^  -^  >/3  -  7  X    *^  i/(4  +  6  2/)8 

14.  re'«+-cZx,    f45-3xfZa;,    T ^ ,     pin  (log x)^^ 

-'  -^  ^  (1  +  x-)tan-ix     J         X 

15.  ^t(t-l)^dt,    (  (a -{- by)^  dy,    ( (m -\- z)^  dz,    fcosfxcZx. 

16.  i  cos^ X  rfx,    I  sec* x  cZx,    i  sin^ x  (?x,    i  sec^  (-\d0. 

--      f   sin  X  dx        r  cos  x  cZx        r     sec^  x  (^x         T      sec^xdx 
J3  +  7cosx'    J9-2sinx'    J  V4  -  3  tan  x     •'  VlO  -  3  sed^' 

18.     f      ^^"^     ,     f(a2_x2)%(?x,     fVc^H:^).^.^^^    f      ^^^     . 
•^Va-^  +  x-^     -^  -^  •^(a2-x-^)t 

106.  Integration  by  parts.  Let  u  and  v  denote  functions  of  a 
variable,  say  x ;  then  [Art.  32  (7)] 

d  (uv)  =  udv  -{-  V  dnj 

whence  u  dv  =  d  (uv)  —  v  du. 

Hence,  on  integration  of  both  members, 

iudv  =  uv  -  \v  du.  (1) 

If  an  expression  f(x)dx  is  not  readily  integrable,  it  may  be 
divided  into  two  factors,  ic  and   dv   say.      The   application   of 

formula  (1)  will  lead  to  the  integral   |  v  du,  and  it  may  happen 

that  this  integral  can  easily  be  found. 

Note  1,  The  method  of  integi-ating  by  the  application  of  formula  (1)  is 
called  integration  by  parts.  This  is  one  of  the  most  important  of  the  par- 
ticidar  methods  of  integration. 


178  INFINITESIMAL    CALCULUS.  [Ch.  XL 

EXAMPLES. 

1.  Find   \  xe*  dx. 

Put  u  =  X',  then  dv  =  e^ dx, 

du  =  dx,  and      v  =  e'. 

.'.  \  xe^  dx  =  xe^  —  I  e^  dx  =  ice^  —  e^  +  c. 

2.  Find  i  sin~i  x  dx. 

Put  t<  =  sin~i  X  ;  then  (?y  =  dx, 

du  =  —    ^      )  and      v  =  x. 

Vl  -x2 

X  dx 


.-.  i  sin-i  xdx  =  X  sin-^  ^  ~  I  " 


Vl  -  x-^ 
=  X  sin-i  X  +  Vl  -x2  +  c.     (See  Ex.  18,  Art.  105.) 

3.  Find  I  x  cos  x  dx. 

Put  M  =  cos  X ;  then  dv  =  x  dx, 

du  =  —  sin  X  dx,  and      v  =  ^x^. 

.'.  J  X  cos  X  dx  =:  I  x2  COS  X  +  ^  ^J^  siu  X  dx. 

Here  the  integral  in  the  second  member  is  not  as  simple  a  form,  from  the 
point  of  view  of  integration,  as  the  given  form  in  the  first  member.  Accord- 
ingly, it  is  necessary  to  try  another  choice  of  the  factors  u  and  dv. 

Put  u=x;  then  dv  =  cos x dx, 

du  =  dx,  and      v  =  sin  x. 

.'.  I  X  cos  X  dx  =  X  sin  X  —  j  sin  x  dx  =  x  sin  x  +  cos  x  +  c. 

4.  Find  i  x^  cos  x  dx. 

Put  u  =  x^  ;  then  dv  =  cos  x  dx, 

du  =  3  x2  dx,  and      v  =  sin  x. 

.'.  j  x^cosxdx  =  x^  sinx  —  3  I  x^sinx  dx.  (1) 

It  is  now  necessary  to  find  i  x^  sin  x  dx. 

Put  ti  =  x^  ;  then  dv  =  sin  x  dx, 

du  =  2x  dx,  and     v  =—  cos  x. 

/.  I  x2  sin  X  dx  =  —  x^  cos  X  +  2  j  X  cos  x  dx.  (2) 


106.]  ELEMENTARY  INTEGRALS,  179 


It  is  now  necessary  to  find  \  x  cos  x  dx. 

By  Ex.  3,  ix  cos  xdx  =  x  sin  x  +  cos  a:  -f  c. 


Substitution  of  this  result  in  (2),  and  then  substitution  of  result  (2)  in 
(1),  gives 

j  x^  cos X  dx  =  x^  sin x  +  3 x-  cos x  —  6xsmx  —  6  cos x  +  Ci. 

When  the  operation  of  integrating  by  parts  has  to  be  performed  several 
times  in  succession,  neatness,  in  arranging  work  is  a  great  aid  in  preventing 
mistakes.    The  work  above  may  be  arranged  much  more  neatly;  thus: 

\  x^  cos  X  c^x  =  x^  sin  x  —  3  I  x^  sin  x  dx 

=  x^  sin  X  —  3    —  x2  cos  X  4-  2  I  X  cos  x  dx 

=  x^sinx  —  3[—  x2cosx+  2(xsinx  +  cosx  +  c)] 
=  x^  sin  X  +  3  x^  cos  x  —  6  x  sin  x  —  6  cos  x  -\-  C 
=  x(x2  -  6)  sin X  +  3(x2  -  2)  cosx  +  C. 
The  subsidiary  work  may  be  kept  in  another  place. 

5.   Find  f  Va^  -  x'^  dx.     (See  Ex.  5,  Art.  105.) 

Put  u  =  Va^  —  x2 ;  then  dv  =  dx, 

du  = ^^^     ,  and      v  =  x. 

Va2  -  x2 

...  C  Va2  _  x^  dx  =  xVa'^-x^  +  (    ^^^^     ■  (1) 

•^  *^  Va^  —  X- 


Kow  va^  —  x^  = 


Va'^  —  x2      Va^  —  x"-^      Va^  —  x^ 


hence  '    — ^^r=  =  — -=.  —  Va^  —  x^. 

Va-^  -  x2      Va2  _  a;2 

Substitution  in  (1)  gives 

f  Va2  _  x^  dx  =  xVd'  -  x2  +  r_^!^__  f  VfT^^fix.  (2) 

Hence,  on  transposition  of  the  last  integral  in  (2)  to  the  first  member, 
division  by  2,  and  Ex.  4,  Art.  105, 

f  Va2  -  x2  dx  =  -  (x  \/a2  -  x^  f  a2  gin-i  ^V 


180  INFINITESIMAL   CALCULUS.  [Ch.  XI. 

6.  I  e^  cos xdx  =  ^  e^  (sin  x  +  cos x). 

(Integrate,  putting  u  =  e"",  then  integrate,  putting  u  =  cosaj.     Take  half 
the  sum  of  the  two  results.) 

7.  xxe'^^'dx.  11.     ixlogxdx.  16.     jx^siniccte. 

8.  ixe-='dx.  12.     \x^\ogxdx.  16.     ie='x'>'dx. 

9.  ix^f^dx.  13.     (tan-ixdx.  .       17.     (  a;  sin  x  cos  x die. 

10.     flogxdx.  14.     fxtan-ixfZx.  18.     f ---^^^  dx. 

J  J  *^  V  i  -  x2 

19.    Derive  \  e^  sinx  dx  =  |  e^  (sin  x  —  cosx).      (See  Ex.  6.) 


107.  Further  elementary  integrals.  A  further  list  of  elementary 
integrals  is  given  here.  They  can  be  verified  by  differentiation. 
Some  of  the  ways  in  which  they  may  be  derived  are  indicated  in 
the  latter  part  of  the  article. 

XV.    I  tan  u  du  -  log  sec  u-\-  c, 
XVI.    (  cot  u  du  -  log  sin  «*  +  c. 
XVII.    (  sec  u  du  =  log  (sec  u  +  tan  u)  +  c, 


=  logtan(|4-|)  +  c. 

XVIII. 

1  cosec  u  du  =  log  tan  ^  +  ^J* 

XIX. 

C      du       _,in-ii^  +  c. 

^  Va2  _  ^2              a 

XX. 

XXI. 

r        ^^        -^sec-i^  +  c. 

J  u  Vw2  -  a2     a           a 

XXII. 

r         du         _           1  ?* 

1                              —  vcis           -r  */• 

.B.    See  Note  1. 

107.] 


ELEMENTARY  INTEGRALS.  181 


XXIV.    ( — — —  =  log  (u  +  Vu^  +  a^)  +  c. 


XXV.  r      ^^^       =  log  («*  +  ^V^  -  a^)  +  c, 


^«*2  _  (ji 


XXVI.   y  ^^a^  -u^du=^(u  Va2  _  ^2  +  «2  gi^-i  ^\  +  c. 

Integral  XXII.  is  also  reducible  to  form  XIX.     For  2  au  —  i*^ 

=  a-  —  (?^  —  a)^,  and  du  =  d{u  —  a); 

•^  V2  «i*  -  t*2     J  Va^  -  (u  -  ay  "* 

Ex.    Show  that  this  result  and  that  in  XXII.  are  equivalent. 
Remarks  on  integrals  XV.  to  XXVI. 

Formulas  XV.,  XVI.     For  derivation,  see  Exs.  6,  7,  Art.  104. 
Formulas  XVII.,  XVIII. 

cosec  u  —  cot  u 


Since       cosec  u  =  cosec  u 


cosec  u  —  cot  u 


C  J        C  -  cosec  ?/  cot  11  4-  cosec2  u  , 

I  cosec  u  du  =  \ —  clu 

J  J  cosec  u  —  cot  u 

^  rd  (cosec  u- cot  tQ  ^  j^g  ^^^gg^  ^^  _  ^Q^.  ^^>j 
J     cosec  u  —  cot  w 


1-^Q^^  =  log ?—  =  log  tan  ^. 


Substitution  of  m  +  -  for  u  in  the  last  two  lines  gives 
f  cosec  (u  +  -^  d?<  =  log  tan  [^^  +  J^ ,  i.e.   f  sec  m  fZit  =  log  tan  (^  +  j )  5 

=  log  I  cosec  (  «+  -)  -cot /"?<+  I'j  I  =  log  (sec  ?<+tan  u). 
There  are  various  methods  of  deriving  XVIL  and  XVIII. 


182  INFINITESIMAL   CALCULUS.  [Ch.  XI. 

Formulas  XIX.,  XX.,  XXI.,  XXII.,  XXIII.     For  derivation, 
see  Exs.  4,  6,  7,  8,  Art.  105,  and  the  following  suggestion : 

Suggestion  :  — =  —  ( | ;  — =  —  ( 1 ) . 

u^  —  a^     2a\u  —  a     u-\-aj    a^  —  u^     2a\a+u     a  —  nj 

Formula  XXIV. 

Put  w2  +  a2  ^  ^2  .  then  udu  =  z  dz,  whence  ^—  =  —  • 

z       u 
Hence,  __Jm_  ^dji^dz^ 

■    v'm2  +  a^      ^       u 

^                 ...               du           du -\- dz     dCii  +  z) 
On  composition,    — :::33:^^  = ■ =  — ^^ — — — ^• 

Vm2  ^  a^        u-Y  z  u  +  z 

.-.    r         ^^  =    f^^-^^^^t^  =  log(M  +  0)   +  C  =  log  (it  +  Vlt^  -h  «2)  +  c. 

•^  Vm2  4  a2     J     If  +  2; 
The  last  result  may  be  written 


log  (u  +  Vm2  +  a^)  -  log  a  +  c',  le.  log  ^^  +  ^^^^  +  ^^  +  c', 

a 

a  form  which  is  convenient  for  some  purposes.     See  Note  3. 

Formula  XXV.  can  be  derived  in  the  same  way  as  XXIV. 

Formula  XXVI.     For  derivation,  see  Ex.  5,  Art.  105,  and  Ex. 
5,  Art.  106. 

Note  1.     Integrals  XIX.,  XX.,  XXL,  XXII.,  may  be  respectively  written 

-  cos-i  -  +  c',    --  cot-i  -  +  c',    -  -  csc-i  -  +  c',   -  covers-i  -  +  c'. 
a  a  a  a  a  a 

Ex.     Show  this. 

:Note  2.     Integrals  XXIII.,  XXIV.,  XXV. ,  may  be  written  thus  : 

f  ^,^  =  -  hy  tan-'  ~  +  c'(ii^  <  a2), 
f-T^  =  --hycot  '''  +  «'(^->«2), 


r^^^  =  ±hycos-i^  +  c'. 
^  Vt*2  _  «2  a 


107.]  ELEMENTARY  INTEGRALS.  183 

The  functions  whose  symbols  are  here  indicated  are  the  inverse  hyperbolic 

tangent  of  -,  the  inverse  hyperbolic  sine  of  -,  and  the  inverse  hyperbolic 

a  a 

cosine  of  —     For  a  note  on  hj'perbolic  functions  see  Appendix,  Note  A. 

a 
The  close  similarity  between  XX.  and  these  forms  of  XXIII.  may  be  remarked ; 
so  also,  between  the  forms  of  XIX.  and  these  forms  of  XXIV.  and  XXV. 

Note  3.  The  same  integral  may  be  obtained  by  various  substitutions,  and 
may  be  expressed  in  a  variety  of  forms.  Instances  of  this  have  already  been 
given  ;  another  example  is  the  following :  Integral  XXIV.  can  also  be  derived 
by  changing  the  variable  from  w  to  2;  by  means  of  the  substitution  y/u"^  +  a'^ 
=  z  —  u:  this  leads  to  the  form 


J: 


^"  log  (u  +  Vm2  +  a2)  +  c. 


Vu^  +  a^ 
The  first  member  can  also  be  integrated  by  changing  the  integral  from  u 


to  z  by  means  of  the  substitution  Vu^  +  a'^  =  zu  ;  this  leads  to  the  form 

y/u^  4-  d^  +  u\h 


^  Vu^  +  a'2  *-  Vu'^  +  a-  —  M  ^ 


It  is  left  as  an  exercise  for  the  student,  to  employ*  these  substitutions  in 
the  integration  of  XXIV.,  and,  the  arbitrary  constants  of  integration  being 
excepted,  to  show  the  identity  of  the  various  forms  obtained  for  the  integral. 

EXAMPLES. 

1.  fi±^dx=f(-A_  +  _L^y:.  =  2tan-i^  +  Ilog(4  +  a:2)+c. 
J  4  +  a;"-^  J  V4  +  x2     4  +  X-  y  2      2 

2.  (±tl^dx=  ff       ^       +-I^Vzx=4sin-ig-7(4-xM+c. 

3      (  (^^  =(     ^(^  +  2)      ^1  tan-i  ^-  +  ^  +  c 

Jx2  +  4x  +  20     J  (x  + 2)2 +  16      4  4 

4«.    C  ^^      _^  r — g(x  +  2)        ^ioc,(-a;+2+\/x2+4x+20)+c. 

•^  Vx2+4a:+20     -^  V(x  +  2)2  +  16 

4?,.    r  ^x  ^r       d(x  +  2)        ^si^-i^±j  +  ,. 

*^  V12  -  x-!  -  4  X      *^  Vie  -  (X  +  2)2  4 

Notice  should  be  taken  of  the  aid  afforded  {e.g.  in  Exs.  3,  4  a,  4  6)  by 
completing  a  square  involving  the  terms  in  x. 


¥-■ 


184 


INFINITESIMAL   CALCULUS. 


[Ch.  XL 


X  = 


dx 


Put  X  = 


Wm-x^ 

1 


Then  dx  = dt,  and 


dx 


V16 


--I 


tdt 


1     ..n.o  .X*     ,      .  (16-X^)^ 


16 


10. 


11. 


12. 

13. 
14. 


r ^ 

J  »;2  +  5  a 


a;2-f  6X+17 

7  —  6  X  —  x^  ' 
dx 


Vie  f2  - 1 

dx 

VlT  +  6x  — X' 


^J(16«2_i)-J^(16«2_i) 


10  X 


^  +  c. 


i^  («)]■ 


dx 


Vx2  +  6x  +  10 
dx 

Vx2  -  5  X  +  7 


r       dx 

J  4  x2  -  5  X  -I  6 


(2)1 

(2)  f-=^i==;    (3)  f: 

•^  V  7  —  O  X  —  x2  •^ 

(2)     r <^ ;      (3)    f  ^^ 

J  x^  +  5 X  -  y  J  \/4x2-8x  + 

(2)  r      ^^  — -;  (3)  r ^ 

-'a/9-5x-4x2  J7-5x 


4x2 


\/8x 


X 

fZx 


dx 


5xV9x2-25 


f ^^^  ;     (2)   f  V9=:^(7x;     (3)  ('V2o-x^dx. 

•^  (x-l)Vx'-2x-3     ^       *^  -^0 

I  VSQ  ~  ix^dx  ;     (2)   J  sec  3  x  dx  ;     (3)   Tcosec  (4  x  —  «)  dx. 

f  tan  (3  X  +  a) dx  ;     (2)   Toot  (4  x2  +  a2)x dx  ;    (3)   ("sec  2  xdx. 
15.    Derive  integrals  62  a,  b,  63  a,  &,  p.  406. 
16. 


•^        ^*  *^  (-4  4.  x2)t     •^xV'12x 


108.   Integration   of   f(x)dx   when  /"(jr)    is  a  rational  fraction. 

In  order  to  find    lf(x)dx  when  f(x)  is  a  rational  fraction,  the 

procedure  is  as  follows : 

Resolve  f(x)  into  component  fractions,  ayid  integrate  the  differ- 
entials involving  the  component  fractions. 

Note.  It  is  here  taken  for  granted  that  in  his  course  in  algebra  the 
student  has  been  made  familiar  with  the  decomposition  of  a  rational  fraction 
into  component  fractions,  or,  as  it  is  usually  termed,  the  resolution  of  a. 
rational  fraction  into  partial  fractions.  Reference  mky  be  made  to  works 
on  algebra,  e.g.  Chrystal,  Algebra,  Part  I.,  Chap.  VIII.  ;  also  to  texts  on 
calculus,  e.g.  Snyder  and  Hutchinson,  Calcithis,  Arts.  132-137. 


108.]  ELEMENTARY  INTEGRALS.  185 

Examples  1,  2,  4  will  serve  to  recall  to  mind  the  practical 
points  that  are  necessary  for  present  purposes. 

EXAMPLES. 

J  x2  +  ic  -  6 

x^-\-  x  —  H  x2  -\-  X  —  6 

The  fraction  in  the  second  member  is  a  proper  fraction,  and  is  in  its 
lowest  terms.  Accordingly,  the  work  of  resolving  it  into  fractions  having 
denominators  of  lower  degree  than  the  second,  may  be  proceeded  with. 
Since  its  denominator,  x^  +  a;  -  tj,  i.e.  (x  —  2)(x  +  3),  is  the  common  denom- 
inator of  the  component  fractions,  one  of  the  latter  evidently  must  have  a 
denominator  x  —  2,  and  the  other  a  denominator  x  +  3,  Since  these  frac- 
tions must  be  proper  fractions,  their  numerators  must  be  of  lower  degrees 
than  the  denominators,  and,  accordingly,  must  be  constants. 
Accordingly,  put 

14  X  -  10       /       14  X  -  10      \         A      ,      B 


Q_  ^  /       14X-10      \  ^     A 

6       \  (x  -  2)  (X  +  3)  /      X  -  2 


x2  +  X  -  6       \  (x  -  2)  (X  +  3)  /      X  -  2  ■  X  -f-  3  '         ^  ^ 

Here  A  and  B  are  to  be  determined  so  that  the  two  members  of  (1)  shall 
be  identically  equal. 

On  clearing  of  fractions, 

14x- 10  =r  ^(x-h  3)-h  J?(x-2).  (2) 

Since  the  members  of  (2)  are  to  be  identically  equal,  the  coefficients  of 
like  powers  of  x  must  be  equal.     That  is, 

^  4-  ^  =  14, 
3  ^  -  2  J5  =  -  10. 

On  solving  these  equations,  A  =  V-,  B  =  Y- 
r,3_3.-2  +  4x+14^^^r/     _  18  52       \^^ 

J  y^i  +  x-Q  J  \  5(x  -  2)      5(x  -f  3)  / 

::,  ^  a-2  -  4  X  +  ^^  log  (x  -  2)  -h  ^V  log  (x  -|-  3)  -}-  c. 
Another  way  of  finding  A  and  B  in  (2)  is  the  following  : 
The  two  members  of  (2)  are  to  be  identically  equal,  and  accordingly  equal 

for  all  values  of  x. 

Now,  put    X  =  —  3  ;   then   —  5  5  =  —  52  ;   whence,  B  =  ^J^. 
Put  x  =  2;        then       5^1  =  18;       whence,  yl  =  V". 

Note  1.     Any  other  values,  e.g.  3  and  7,  may  be  assigned  to  x  ;  in  this 

case,  however,  the  values  2  and  —  3  give  the  most  convenient  equations  for 

determining  A  and  B. 

Note  2.     For  a  more  rapid  way  of  finding  A  and  B  in  such  cases  as  (1), 

see  Murray,  Integral  Calculus,  Appendix,  Note  A. 


186  INFINITESIMAL   CALCULUS.  [Ch.  XL 

X-  +  21:>:  -  10 


J 


x^  +  x^  —  5  X  +  3 


The  fraction  in  the  integrand  is  a  proper  fraction,  and  is  in  its  lowest 
terms.  Accordingly,  the  work  of  decomposing  it  into  fractions  having  de- 
nominators of  degrees  lower  than  the  third  may  be  proceeded  with.  Since 
the  denominator  x^  +  x^  —  5  x  +  3,  i.e.  (x  —  1)^  (x  +  3)  is  the  common 
denominator  of  the  component  fractions,  one  of  the  latter  evidently  must 
have  a  denominator  x  +  3,  and  another  must  have  a  denominator  (x  —  l)^. 
It  is  also  possible  that  there  may  be  a  component  fraction  having  the  denom- 
inator X  —  1 ;  for,  if  there  is  such  a  fraction,  it  does  not  affect  the  given 
common  denominator.     Accordingly,  put 

x2  +  21 X  -  10   „     ^      ,        B        ,      C  ,oN 

+  77 — :r:Ti  +  - — V  KP) 


(x  -  l)-^(x  +  3)     X  -h  3     (X  -  1)2 

in  which  A.,  B,  C  are  constants  to  be  determined. 

On  clearing  of  fractions,  equating  like  powers  of  x  (for  reasons  indicated 
in  Ex.  1),  and  solving  for  A,  i?,  O,  it  is  found  that 


A=-i,     B  =  'S,     C=b. 

r  X3  +  21X-10  ^^^r/^-£         3         ^\ 

J  x3  +  x'^  -  5  X  -i-  3  J  \  X  4-  3      (x  -  1)2     X  -  1  / 


dx 


=  51og(x-l)-41og(x  +  3)--^+c=:log<^i-:4T5-^+c. 

X  —  1  (x  +  3)4      X  —  1 

Note  3.  It  may  be  asked  why  the  numerator  assigned  to  the  quadratic 
denominator  (x  —  1)2  in  the  second  member  of  (3)  is  not  an  expression  of 
the  first  degree  in  x,  say  Bx  +  7>,  instead  of  a  constant.  The  reason  is,  that 
if  such  a  numerator  were  assigned,  the  fraction  would  immediately  reduce  to 
the  forms  in  (3).     For 

Bx  +  D  ^  jB(x-  1)+  Z)+  i?  ^     B  D  +  B 

(X-l)2  (X-l)2  X-1         (X-l)2' 

forms  which  appear  in  (3). 

Note  4.  If  a  factor  of  the  form  (x  —  a)'"  appears  among  the  factors  of  the 
denominator  of  the  fraction  to  be  resolved,  there  evidently  must  be  a  com- 
ponent fraction  having  (x  —  ay  for  its  denominator.  There  may  also  possi- 
bly be  fractions  having  as  denominators  (x  —  a)  of  various  powers  less  than 
r,  e.g.  (x  —  a)''"^,  (x  —  a)''-2,  ...,  x  —  a.  Accordingly,  in  such  a  case  it  is 
necessary  to  allow  also  for  the  possibility  of  the  existence  of  fractions  of  the 

M  F  L 


(x-ay-^     (x-ay-^ 
in  which  M,  F,  •••,  L,  are  constants. 


108.]  ELEMENTARY  INTEGRALS.  187 

J2  a;2  —  8  a;  —  10 
dx.     (Compare  denominators  in  Exs.  2,  3.) 

4.     (-5x^  +  3. +  17    ^^ 
J  x3  -  x=^  +  4  X  -  4 

The  fraction  in  the  integrand  is  a  proper  fraction  and  is  in  its  lowest  terms. 
If  it  were  not  so,  division  as  in  Ex.  1  and  reduction  would  be  necessary. 
Since  the  denominator  x^  —  x-  +  4 x  —  4,  i.e.  (x^  +  4)(x  —  1),  is  the  com- 
mon denominator  of  the  component  fractions,  one  of  the  latter  must  have  a 
denominator  x^  +  4,  and  the  other  a  denominator  x  —  1.     Accordingly,  put 

6x2  4-  3  X  +  17  _Ax-]-  B  .      C 

1  r »    . 


(x-^  +  4)(x-l)       x2  +  4 

in  which  A,  B,  C,  are  constants  to  be  determined. 

On  clearing  of  fractions,  equating  coefficients  of  like  powers  of  x,  and 
solving  for  A,  jB,  C,  it  is  found  that 

A  =  0,  B  =  S,  C  =  5. 

._  r  5x^  +  3x4-17  ^^^r/__3_  +  -^Ux 

Jx3-x2  +  4x-4  JVx2  +  4     x-iy 

=  |tan-if+51og(x-l)+c. 

Note  5.  The  expression  x^  +  4  has  factors  x  +  2  i,  x  —  2  i  (i  —  V—  1) ; 
if  these  be  taken,  component  fractions  imaginary  in  form,  are  obtained.  It 
is  usual,  however,  not  to  carry  the  decomposition  of  a  fraction  as  far  as  the 
stage  in  which  component  fractions  imaginary  in  form  may  appear. 

Note  6.  The  numerator  Ax  +  J5  is  assigned  above  ;  for  the  numerator 
over  a  quadratic  denominator  whose  factors  are  imaginary,  may  have  the 
form  of  the  most  general  expression  of  the  first  degree  in  x. 

Note  7.  When  a  quadratic  expression  x^  -\-  px  -\-  q  has  imaginary  factors 
and  is  repeated  r  times  in  the  denominator  of  a  fraction,  in  the  process  of 
decomposition  of  this  fraction  allowance  must  be  made  for  fractions  of  the 

forms,  ^^  +  ^      ,    __Cx  +  D ..        Mx  +  N    , 

{x^  +  px  +  qy.    (x:' +  px  +  q)'-'  x'' +  px  +  q 

5.    (1)    rn  x^-4x  +  28^  r.s^2_i3x-5    ^^      (Compare  the 

^^Jx3-x-^  +  4x-4      '^^Ja;3-x2  +  4x-4  ^ 

denominators  in  Exs.  4,  6.) 


188  INFINITESIMAL    CALCULUS.  [Ch.  XI. 

Find  the  anti-derivatives  of  the  following  fractions  : 


12. 


a;  +  37 

xi  _  ;j  a;  _  28 

8x+  1 

2  x-^  -  9  X  —  35 

X3  _  2  X2  -  1 

X-^-1 

x'^  -  a:2  -H  1 

x^-x 

a:2_i0x-5 

a;(2  .x^  +  3  a;  -  5) 

x2+pg 

x(x-p)(x  +  (/) 

Ilx3-llx2-74x 

+  84 

6. •^"^''' 17. 

x-2  _  3  a;  _  28 

_   8X+1 

2  x^  -  9  X  —  35 

8.  ^^-^^^-^  19. 
x^  -  1 

9.  ?^^1  +  1.  20. 

X^  —  X 

10.       a:2--10x-5  ^  2^ 

11. 


.    2  x2 

x2  -  3  X  +  3 


X*  -  13  x2  + 


^3-    /"^.\./  24. 

(X  -  1)^ 

14.  ^^  +  ^    .  25. 

(4x+5)2 

15.  5a;2  +  x-10^  26. 


x(x2  +  3) 

12  -  X  -  x2 

(3x-2)(x2  +  5) 

(X+1)2 

x^  +  x 

X3-1 

x3  +  3x 

2  x2  +  3  X  +  6 

x3  +  3  X 

7x2  +  9 

x«  +  3x 

2  x3  -  x2  +  8  X  +  12 

x-^(x2  +  4) 

2  +  3  X  -  x2 

(x-l)(x2-2x+5) 

1  +  7  X  +  x"  +  x3 

16. 


x2(2  X  +  5) 
30x2  +  43x-8 


(x  +  4)(3x  +  2)2 

Ex.  27.  Show  that  any  expression  of  the  form  f  0^^^  +  n)dx^  .^  which 
w,  n,  a,  6,  and  c  are  constants,  is  integrable.  ^       ^ 

109.  Integration  of  a  total  differential.  In  Art.  86  it  has  been 
shown  that  the  necessary  condition  for  the  existence  of  a  function 

^'^""^  Pd^^qay  (1) 

for  its  differential,  is  that     -r-  =  ^»  (2) 

'  dy     doc  ^  ^ 

It  has  also  been  stated  (Art.  86,  Note  1)  that  condition  (2)  is 
sufficient  for  the  existence  of  such  a  function.  In  other  words, 
if  the  expression  (1)  has  an  anti-differential  (or  integral),  relation 
(2)  must  be  satisfied ;  conversely,  if  relation  (2)  is  satisfied,  the 
expression  (1)  has  an  integral.  Accordingly,  relation  (2)  is  called 
the  criterion  of  integrability  for  the  expression  (1).    If  this  criterion 


109.]  ELEMENTARY  INTEGRALS,  189 

is  satisfied,  the  expression  (1)  is  said  to  be  a  coynplete  differential^ 

a  total  differential,  and  also  an  exact  differential. 

If  test  (2)  is  satisfied,  the  integral  of  (1)  can  easily  be  found. 

This  integral's   partial  ^-differential,  Pdx,  can  only  come  from 

terms  containing  x  (Art.  79).     Hence,  the  integral  of  Pdx  with 

respect  to  x,  namely,  "    /* 

jPdx  +  c,  (3) 

must  yield  all  the  terms  of  the  required  integral  that  contain  x. 
Also,  Qdy  can  only  come  from  terms  containing  y.  Hence  the 
integral  of  ^  dy  with  respect  to  y,  namely. 


/ 


Qdy  +  c,  (4) 

must  yield  all  the  terms  of  the  required  integral  that  contain  y. 
Some  of  these  terms  mg,y  contain  x ;  if  so,  they  have  already  been 
obtained  in  (3),  and  need  not  be  taken  this  second  time.  Hence, 
if  the  integral  of  a  differential  of  the  form 

Pdx+Qdy 

is  required,  apply  the  test  for  integrability,  namely, 

dP^dQ^ 
dy      dx ' 

if  this  test  is  satisfied,  integrate  Pdx  tvith  respect  to  x  ;  then  integrate 
Qdy  with  respect  to  y,  neglecting  terms  already  obtained  in  |  Pdx ; 
add  the  results  and  the  arbitrary  constant  of  integration. 

EXAMPLES. 

1.  Integrate  (2xy  -^2 +  ?>}/- +  12  x)  dx  +  (a:2  +  6  x?/  +  4  y^)  dy. 
Here  P  =2xy  +  2 -\- Zy'^  -^  \2x,  and     Q  =  x^  +  Qxy  +  ^y^. 

.'.  ^=2x-\-6y,  and  ^  =  2  x-\-6y. 

dy  dx 

Thus  the  criterion  of  integrab'ility  is  satisfied. 
Also   (pdx  =  x'^y  -\- 2  X  +  S  xy^ -{■  Q  x"^  ] 

and  I  Qdy  =  x^y  -\-  S  xy^  +  y*,  in  which  y*  has  not  been  already  obtained 
in    (  Pdx.     Hence  the  integral  is 

a:2y  4-  2  a:  +  3  y%j-  6x'^  +  y^-\-c. 


190  INFINITESIMAL   CALCULUS.  [Ch.  XI. 

2.  Verify  the  result  in  Ex.  1  by  differentiation. 

3.  Find   \(xdy  —  ydx). 

Here  ^  =  1,  and  ^  =  —  1 ;  hence  the  test  for  integrability  is  not  satis- 
dx  dy 

fied,  and  there  is  not  an  anti-differential. 

4.  (1)  ( e^  (cos ydx- sin  ydy).    (2)  C [(Sx'^+8xy-\-i)dx-\-iix^-Q)dy;\. 

5.  Integrate  :  (1)  cos  x  sec^  y  dy  —  (sin  x  tan  y  +  cos  a;)  dx. 

(2)  (xey  -2x)dy  -\-  (ey  -2y  -\-2x) dx.     (3)   (S  -  ix  -  y)dx  -  (x -{■  y) dy. 

N.B.  An  accurate  and  ready  memory  of  the  fundamental  inte- 
grals (Arts.  103,  107),  resourcefulness  in  making  substitutions 
(Art.  105),  and  quickness  in  integrating  by  parts  (Art.  106),  are 
three  very  important  things  to  cultivate  in  order  to  insure  com- 
fortable progress  in  the  study  of  the  calculus. 

EXAMPLES. 

1.  ( ln^x'^+"' dx,   f  (a  + ft)a:2(«+6)-idfx,    ( (r  +  s)z^+*+'^dz,   ( rh^y'-'^  dy, 

rt^^^ei±l  r.3  +  8.'^-9  r^^^^2x^  +  7x-l  C     1  dt    ^ 

Jo        t  +  2  J       t?2  +  3  J  x?--2  J  9  ^-^  4-20 

r^^,     i\^^y^ydy,    f^^^,    r^^^,    f    ^'-^     dz. 
Jz^-\2     Ji  ^      ^  ^  ^     ^'    J  V9^::^6     J  Vx6"~9     J  (2  ^  -  1)2 

TT 

2.  Ttan  (ma;  +  w)  (7x,    f  (sec  3  a;  +  2)2dx,    f   tan2^(Z^,    psin/- +  ^"J  (Z^. 

3.  icos~ia;da;,  isec-ix(?a;,  xcoAr^xdx,  j(logx)2c?x,  I »;%  «cZa;, 
j  x^e-^  dx,    i  sin  x  log  cos  x,    j  x"*  log  x. 

4.  f^Lziclx,    fJ^dx,    f'-*^,     fjiil<to. 

Jo  2         Jo  e3x     Jo  Ji        ^1  _  ^2 

g    r_asin^_^  r(l  +  cose)dd^        C         dx                  rsecxtanxdx 

J  m  +  n  cos  d  J          sin  ^                J  sin  x  +  cos  x        J  (tan^x  —  3)^ 

dd ^  riog2  (mz  +  n)  ^^          f        dx                C      dx 

cos2  eV4-  tan2^'  ^       mz  +  n          '       J  VoF'^^W^        J  e' +  e-*' 


109.]  ELEMENTARY  INTEGRALS.  191 


sill  -  -v/cos - 
4>'       2 


sill  - 

r     dx  r         dd  c 2 

J  e2x  _  g-2x'  J  cos2 26-  siiV^ 2 ^'  J    .    a:    / 

sin^^ 

4  [(1  —  sin  a; oos y)  dx  —  (cos xsmy  ■\-2y)  c??/], 
j  [(1  —  sinajsini/)  dx  +  (cos a; cosy  —  1)  dy']. 

7.  Derive  the  following  integrals  : 

(1)  fx(x2±a2rda:  =  (^i±-«^l^.  (2)  f-^^^  =  V^^T^ 

*^  2(H+1)  'J  Vx--^±a--^ 

(3)  f  a:(a2  _  x^ydx  =  -  (<^' -  ^'y'^\        (4)  C_^dx_  ^ _  Va^^^. 
^^  2(?i  +  l)  'J  Va2"^^^ 

8.  Derive  the  following  integrals  : 

(1)  f-^  =  ^log(a  +  6x).     (2)  r(a4-6x)»dx  =  -^^i^^,  whenn 
J  a  +  bx      b  J  b{n  +  1) 

is  different  from  -  1.  (3)  C  ^^^    =lra-\-bx-a  log  (a  +  6x)]. 

J  a  -{-  bx     b" 

(^)  r-^!^  =  l[i(a  +  &:^)^-2a(«  +  6x)  +  a-21og(a  +  6a:)].  (5)  f   /^ 
J  a  +  bx     b^  J  x{a  +  bx) 

=  --Mog^  +  K     (6)  f ^^ =  -i-+Aiog^L+^.     (7)  r     ^^^ 

=  ^Jlog(a  +  6a;)+— ^]. 
62  L  a  +  6a;  J 

9.  Derive  the  following  integrals  : 

(1)  r_i??_=A_iog^±-^.     (2)   f     ^^      ^-l^tan-ia^-J^when 
^'Ja2-6%2     2a6     °  a  -  6a;      ^  ^  J  a  +  6x2      ^^  \a' 

a>0and6>0.        (3)  f    ^^^    ^  J- log  f  a;^  +  "  V        i.^)  f-^i^^?- 
^  ^  Ja  + 6a;2     2  6     ''V         6/  ^  ^  J  a  +  6a;2     6 

gf     ^^     .      (5)  f ^ ^J-log       ^'      .     (6)  f ^ =  -1- 

bJ  a^bx'^       ^  ^  J  x{a  +  6a;2)      2  a     ''  a  +  6a;2      ^  '  J  x'\a  +  6x2)  ^x 

_b  C    dx         ,^.   C      xdx 1 

aJ  a  +  bx^'    ^^J(a+6x2)"~     2  6(n -l)(a  +  6x2)'»-i' 

10.    Derive  the  following  integrals  : 

(1)  Jx  V^TF^  dx  ^  _  2(2  g  -  3  6x)  A/(a  +  hx)\     ^2)  ^x^V^^Tbidx  = 

2(8  a2-12  g  6X+15  62x2)  \/(a  +  6x)3  r    x(Zx     ^     2(2  q-6x)  ^/— -^ 

105  63  ■      ^  ^  J  V^+6i  -^  ^' 

(4)  r_^i^  =  2(8«2-4«6x  +  362x2)  ^^^^  ^.^  ('        ^^        = 
•^  Va  +  6x                      15  63                                J  X  Va  +  6x 

1    Jqo.  Va  +  6x-\/a 
Va        Va 


ij;-^,  for  g>0;  _A^ tan'i J^i  +  ^,  for  a<0. 
+  6x  +  Va  V-  a  ^    -a 


CHAPTER   XII. 

SIMPLE   GEOMETRICAL   APPLICATIONS   OF 
INTEGRATION. 

110.  This  chapter  treats  of  some  simple  geometrical  applica- 
tioDS  of  integration.  Examples  of  some  of  these  applications 
have  already  appeared  in  Arts.  96,  97.  In  Art.  Ill  integration  is 
used  in  measuring  plane  areas,  in  Art.  112  in  measuring  the 
volumes  of  solids  of  revolution.  In  Art.  113  the  equations  of 
curves  are  deduced  from  given  properties  whose  expression  involves 
derivatives  or  differentials. 

N.B.  The  student  is  strongly  recommended  to  draw  the  figure  for  each 
example.  In  the  case  of  examples  which  are  solved  in  the  text  he  will  find 
it  extremely  beneficial  to  solve,  or  try  to  solve,  the  examples  independently 
of  the  book. 

111.  Areas  of  curves :  Cartesian  coordinates. 

A,  Rectangular  axes.  In  Art.  96  it  has  been  shown  that  for  a 
figure  bounded  by  the  curve 

the  a7-axis,  and  the  two  ordinates  for  which  x  =  a  and  x  =  h  respec- 
tively, the  axes  being  rectangular,  area  of  figure  =  limit  of  sum  of 
quantities  y  \x  (or  f(x)  Ax)  when  Ax  approaches  zero  and  x  varies 

continuously  from  a  to  b.     This  limit  is  denoted  by    I    ydx  or 

f{x)  dx',  it  is  obtained  by  finding  the  anti-differential  of /(a;)  dx, 

substituting  b  and  a  in  turn  for  x  in  this  anti-differential,  and 
taking  the  difference  between  the  results  of  the  substitutions. 
In  fewer  words :  the  7iumber  of  units  in  the  area  is  the  same  as  the 
number  of  units  in  a  certain  definite  integral;  namely, 

area  of  figure  =  f   ydie=\    f{x)  dx,  (X) 

Ja  Ja 

The  infinitesimal  differential  y  dx  is  called  an  element  of  area. 

192 


110,111.] 


AREAS  OF  CURVES. 


193 


N.B.     It  will  be  found  that  in  many  problems  it  is  necessary  : 

(1)  To  find  a  differential  expression  for  an  infinitesimal  element  of  area, 
or  volume,  or  length,  etc.,  as  the  case  may  be. 

(2)  To  reduce  this  expression  to  another  involving  only  a  single  variable. 

(3)  To  integrate  the  second  expression  between  limits  (end-values  of  the 
variable),  which  are  either  assigned  or  determinable. 

B,  Oblique  axes.  Suppose  that  the  axes  are  inclined  at  an 
angle  w,  and  that  the  area  of  the 
figure  bounded  by  the  curve  whose 
equation  is  y=f(x),  the  x-axis,  and 
the  ordiuates  AP  and  BQ  (for  which 
x  =  a  and  x=  b  respectively),  is 
required.  Let  BM  be  a  parallelo- 
gram inscribed  between  A  and  B,  as 
rectangles  were  inscribed  in  the 
figures  in  Arts.  95,  96. 

Area  of  RM=  yAx  •  sin  w. 

Area  APQB  =  limit  of  sum  of  all  the  parallelograms  like 
RM,  infinite  in  number,  that  can  be  inscribed  between  AP  and 
BQ ;  that  is, 

Xx=b  /*b 

y  sin  (o  •  cZa;  =  sin  «  I    y  dx. 


area 


Unless  otherwise  specified,  the  axes  used  in  the  examples  in 
this  chapter  are  rectangular. 


EXAMPLES. 

1.   Find  the  area  between  the  line  2^/— 5a;  —  7  =  0,  the  cc-axis,  and  the 
ordinates  for  which  x  =  2  and  a;  =  5. 

The  rectangle  PM  represents  an  element  of  area,  y  dx. 
The  area  required  is  the  limit  of  the  sum  of  these  element- 
ary rectangles,  infinite  in  number,  Jrora  AB  to  DC. 
That  is, 

area  =  T^^  dx  =  \  C {bx  ^1)dx  =  \  1"—+  7  xT 

=  36|  square  units. 

the  unit  of   length   used   in    drawing   the    figure 

one  inch,   the   figure  would    contain    36f    square 

Fig.  48. 


194 


INFINITE^lMA  L   CAL  C  UL  US. 


[Ch.  XII. 


2.   Solve  Ex.  1  without  the  calcuhis,  and  thus  verify  the  result  obtained  by 

the  calculus. 

3.  (a)  Find  the  area  of  the  circle 
a;2  +  2/2  =  9  ;  (6)  find  the  area  of  the  figure 
bounded  by  this  circle,  and  the  chords  for 
which  X  =  1  and  x  ~  2. 

Let  APB  be  the  circle  whose  equation 
is  x2  +  y2  —  9^  Take  a  rectangle  PM,  sup- 
posed to  be  infinitesimal,  with  a  width  dx, 
for  the  element  of  area.  Its  area  is  ydx. 
The  area  of  the  quadrant  AOB  is  the  limit 
of  the  sum  of  all  these  elements  of  area, 
infinite  in  number,  between  0  and  A. 
Hence, 

OAB  =  (''~^ydx=  CVd^'^dx  =  i  fxVO  -  x^  +  9sin-i-1^=  Itt  sq. units. 
Jx=o  Jo  L  3  Jo 

.-.  area  circle  =  4  •  OAB  =  9ir  square  units. 

(6)  Draw  the  ordinates  TE  and  JVL  at  the  points  T  and  N  where  x  =  1 
and  X  =  2  respectively.  The  area  of  TBLN  is  equal  to  the  limit  of  the  sum 
of  all  the  elements  of  area,  PM,  that  lie  between  TB  and  NL.     That  is. 


area  TBLN=C  \jdx  =  f  Vo  -  x^dx  =  i  fxVO  -  x^  +  9  sin-i^]^ 
=  i{  (2  V5  +  9  sin-i  |)  -  (  VS  +  9  sin-i  i)  } 


=  V5-  V2  +  |(sin-i| 


sin-ii). 


Here  the  radian  measures  of  the  angles  are  to  be  employed. 
Now 
V2  =  1.414  ;  sin-i|  =  (41° 40.8')  =  .727  radians  ;  sin-i 


.340  radians. 


.-.  area  required  =  2  .  TBLN  =  5.126  square  units. 

Note  1.     Other  end-values  of  x  may  be  used  in  finding  the  area  of  this 
circle.     Thus 

area  circle  =  2,AiBA  =  2^  ydx  =  2C  VO-x'^dx  =  Tx  V9^^  +  9  sin-i|l 


9sin-il  -9sin-i(-l)  = 


9- 


(-i)- 


IT  square  units. 


Note  2.  These  problems  may  be  stated  thus :  Find  by  the  calculus  (a)  the 
area  of  a  circle  of  radius  3,  (&)  the  area  of  a  segment  between  two  parallel 
chords,  distant  1  and  2  units,  respectively,  from  the  centre.  In  this  case  it 
is  necessary  to  choose  axes  (as  conveniently  as  possible) ,  to  find  the  equation 
of  the  circle,  and  then  to  proceed  as  above. 


111.] 


AREAS   OF  CURVES. 


196 


4.    Find  the  area  between  the  curve  y  =  2x^,  the  y-axis,  and  the  lines 
y  =  2  and  y  =  i. 

The  area  is  represented  by  ABLE.  At  any  point 
P(x,  y)  on  the  arc  BL  take  for  the  element  of  area  an 
infinitesimal  rectangle  3IF.     Its  area  is  x  dy. 

xdy  =—^  \  y^dy 

=  §  .  i_  .  2^  (2^  -  1)  =  ?  (  ^/16  -  1)  =  2.2797. 
4    2^  ^ 


Fig.  50. 


Note  3.  The  definite  integral  which  gives  the  area  may  also  be  expressed 
in  terms  of  x.  For,  since  y  =  2x^,  dy  =  6x^dx ;  also,  when  y  =  2,  x  =  l, 
and  when  y  =  4,  x  —  v^. 

.-.  area  ABLE  =  ("^\  dy  =  (^^^6  x^dx=U  v^  -  1)  =  2.2797. 

6.  (a)  Find  the  area  of  the  figure  bounded  by  the  parabola  y^  =  i  ax, 
the  X-axis,  and  the  ordinate  for  which  x  =  xi.  Show  that  this  area  is  equal 
to  two-thirds  of  the  rectangle  circumscribing  the  figure.  (6)  Find  the  area 
bounded  by  the  parabola  y^  =  9x,  and  the  chords  for  which  x  =  4  and 
x  =  9. 

6.  Find  the  area  between  the  curve  y^  =  4:X^  the  axis  of  y,  and  the  line 
whose  equation  is  y  =  6. 

7.  Find  the  area  included  between  the  parabolas  whose  equations  are 

1/2  =  8x  and  x'^  =  Sy. 
^  The  parabolas  are  OML  and  OBL  ;  the  area  of 

OBLMO  is  required.     To  find  the  points  of  inter- 
y^*     section  of  the  curve,  solve  these  equations  sinml- 
taneously.    This  gives  (0,  0)  the  point  O,  which 
is  otherwise  apparent,  and  (8,  8)  the  point  L. 

Area  OBLMO  =  area  OBLN -  area  OMLN 
=  V8j  x^dx-\\^x'^dx 
=  H^~  —  ¥  =  21|  square  units. 


Fig.  51. 


8.  Find  the  area  included  between  the  parabolas  whose  equations  are 
Sy^  =  2b3>  and  5  x.^  =  9  y. 


196 


INFINITESIMAL   CALCULUS. 


[Ch.  XII. 


9.    Find  the  area  included  between  the  parabola   (y  —  x  —  3)2  =  x,  the 
axes  of  coordinates,  and  the  line  x  =  9.     Figure  52  shows  that  this  problem 
is  ambiguous,  for  OTGML  and  OTKNL  are  each 
bounded  as  described.      On  solving  the  equation  of 
the  curve  for  y, 

y  -^x  ±  Vx  +  3. 

Thus  if         Oq^x,   qG  =  x  +  y/x-^  3, 

and  QK  =  x  —  Vx  +  3. 

.-.  area  OTGML 


\'  {x  -\-  Vx  -f  3)  dx  =  85|  square  units 


and  Sivea.  OTKNL 


\    (x  —  Vx  +  3)  dx  =  49|  square  units. 


Also,  the  area  MTN  (the  figure  bounded  by  the 
curve  and  the  chord  for  which  x  =  9)  =  area,  OTGML  —  a,rea,  OTKNL 
=  36  square  units. 

The  area  of  MTN  can  also  be  found  as  follows  : 

Area  MTN  —  limit  of  sum  of  infinite  number  of  infinitesimal  strips,  like 
KG^  lying  between  T  and  MN. 

Now  strip  KG  =  {QG  -  QK)  dx  =  2  Vx dx. 


area  MTN 


=  C^^xdx 


36. 


10.  Apply  the  second  method  used  in  finding  area  MTN  in  Ex.  9  to  find- 
ing the  areas  in  Exs.  7  and  8. 

11.  Find  in  two  ways  the  area  between  the  parabola  (y  —  x  —  5)^  =  x  and 
the  chord  for  which  x  =  5. 

12.  Find  the  area  between  the  parabola  y  =  x^  —  Sx 
-\- 12,  the  X-axis,  and  the  ordinates  at  x  =  1  and  x  =  9. 

Area  =  P~%7  dx  =  C{x'^-Sx-^  12)  dx 

=  18|  square  units.  (1) 

The  parabola  crosses  the  x-axis  at  B  and  C  where 

X  =  2  and  x  =  6. 

Area  APB  =  P~%  dx  =  24 ; 

area  BEC  =Cydx  =  -  lOf  ; 

^re^CQD=j^ydx  =  27.  ^^^   .^ 


111.] 


ABEAS  OF  CURVES. 


197 


Area  required  =  area  APB  +  area  BEC  +  area  CQD 

=  2i  -  lOf-  +  27  =  18f,  as  in  (1). 

The  sign  of  the  area  BEC  comes  out  negative,  because  the  element  of  area, 
y  dx,  is  negative  as  x  increases  from  OB  to  OC ;  for  dx  is  then  positive  and  y 
is  negative.  On  the  other  hand  as  x  proceeds  from  Ato  B  and  from  C  to  Z), 
y  dx  is  positive.  The  actual  area  shaded  in  the  figure  is  2^  +  10|  -f  27,  i.e. 
40  square  units. 

N.B.  It  should  be  carefully  observed,  as  illustrated  in  this  example,  that 
in  the  calculus  method  of  finding  areas  bounded  by  a  curve,  the  a;-axis,  and 
a  pair  of  ordinates,  areas  above  the  x-axis  come  out  with  a  positive,  and  areas 
belov^r  the  .K-axis  come  out  with  a  negative  sign.  Accordingly,  the  calculus 
gives  the  algebraic  sum  of  these  areas  ;  and  this  is  really  the  difference  between 
the  areas  above  the  x-axis  and  the  areas  below  it. 

13.  (a)  Find  the  area  bounded  by  the  x-axis  and  a  semi-undulation  of 
the  sine  curve  y  =  sin  2x.  (6)  Find  the  area  bounded  by  the  x-axis  and  a 
complete  undulation  of  the  same  curve,  (c)  Explain  the  result  zei'o  which 
the  calculus  gives  for  (6).  (d)  What  is  the  number  of  square  units  bounded 
as  in  (&) ? 

14.  Construct  the  figure,  and  show  that,  according  to  the  calculus  method 
of  computing  areas,  the  area  between  the  curve  whose  equation  is  12  y=(x  —  l) 
(x  —  3)  (x  —  5),  the  X-axis,  and  the  ordinates  for  which  x  =  —  2  and  x  =  7,  is 
—  fl  square  units;  but  that  the 
actual  number  of  square  units  in 
the  figure  thus  bounded  is  12||. 

15.  Find  the  area  between  the 
line  2?/  —  5x  —  7  =  0,  the  x-axis, 
and  the  ordinates  for  which  x  =  2 
and  X  =  5,  the  axes  being  inclined 
at  an  angle  60°. 

Area  APQB  =  ('^y  sin  60°  •  dx 

=  sin  60°  p(5x-l-7)dx 
=  63.65  square  units. 

Note  4.  In  the  light  of  the 
preceding  examples  attention  may 
be    again    directed    to    the    N.B. 

above.  These  examples  also  show:  (1)  the  element  of  area  may  be 
chosen  in  various  ways  (compare  Exs.  1,  4,  7,  9,  11)  ;  (2)  the  end  values 
used  in  a  problem  may  be  chosen  in  different  ways  (see  Ex.  3,  Note  1); 
(3)  the  calculas  method  of  computing  areas  should  not  be  employed  in  a  rule 
of  thumb  way,  but  with  understanding  and  discretiMi  (see  Exs,  12,  13,  14). 


198  INFINITESIMAL   CALCULUS.  [Ch.  XII. 

Note  5.  Precautions  to  be  taken  in  finding  areas  and  computing 
integrals.  Suppose  that  the  area  bounded  by  the  curve  y=f(x),  the  x- 
axis,  and  the  ordinates  at  A  and  B  for  which  x  =  a  and  x  =  h  respectively, 
is  required.  If  the  curve  has  an  infinite  ordinate  between  A  and  B,  or  if 
the  ordinate  is  infinite  at  A  or  B,  or  at  both  A  and  B,  or  if  either  or  both 
the  end  values  a  and  b  are  infinite,  the  area  may  be  finite  or  it  may  be  infinite. 
It  all  depends  on  the  curve  ;  in  one  curve  the  area  may  be  finite,  in  another 
curve  it  may  be  infinite.  When  infinite  ordinates  occur,  either  within  or 
bounding  the  area  whose  measure  is  required,  and  also  when  the  end-values 
are  infinite,  special  care  is  necessary  in  applying  the  calculus  to  compute  the 
area.  The  calculus  method  for  finding  areas  and  evaluating  definite  integrals 
can  be  used  immediately  with  full  confidence,  only  when  the  end  values  a 
and  h  are  finite  a.nd  when  there  is  no  infinite  ordinate  for  any  value  of  x  from 
a  to  6  inclusive.  For  illustrations  showing  the  necessity  for  caution  and 
special  investigation  in  other  cases  see  Murray's  Integral  Calculus,  Art.  28, 
Exs.  3,  4,  5,  6,  Art.  29  ;  Gibson,  Calculus,  §  126 ;  Snyder  and  Hutchinson, 
Calculus,  Arts.  152,  155. 

Note  6.  For  the  determination  of  the  areas  of  curves  whose  equations 
are  given  in  polar  coordinates,  see  Art.  136.  The  beginner  is  able  to  proceed 
to  Art.  136  now. 

EXAMPLES. 

16.  Calculate  the  actual  increases  in  area  described  in  the  Note  and  in 
Exs.  2,  4,  Art.  67. 

17.  Find  the  areas  of  the  figures  which  have  the  following  boundaries  : 
(1)  The  curve  y  =  x^  and  the  line  4y  =  x.  (2)  The  parabola  y^  -\-  Sx  and 
the  line  x  +  y  =  0.  (3)  The  semi-cubical  parabola  y^  =  x^  and  the  line 
y  =  2x.  (4)  The  curves  y^  =  x^  and  x^  =  4 y.  (5)  The  axes  and  the  parab- 
ola Vx  -\-  y/y  =  Va.  (6)  The  curve  x"^  +  6y  =  0  and  the  line  y  -f-  3  =  0. 
(7)  The  curve  (y  -\-  4)2  +  (x  -|-  3)2  =  0  and  the  line  ic  -f-  6  =  0.  (8)  The 
hyperbola  xy  =  1  and  the  ordinates  :  (a)  at  x  =  1,  x  =  7  ;  (h)  at  x  —  1, 
X  =  15  ;  (c)  at  X  =  1  and  x  =  n.  (d)  The  hyperbola  xy  =  k^  and  the  ordi- 
nates at  X  =  a  and  x  =  6.     (And  the  x-axis  in  each  case.) 

18.  Find  the  area  of  the  loop  of  the  curve  Sy^  =  x4(3  4-  x). 

19.  Show  that  the  area  of  the  figure  bounded  by  an  arc  of  a  parabola  and 
its  chord  is  two-thirds  the  area  of  a  parallelogram,  two  of  whose  opposite 
sides  are  the  chord  and  a  segment  of  a  tangent  to  the  parabola. 

[Suggestion  :  First  take  a  parallelogram  whose  other  sides  are  parallel  to 
the  axis  of  the  parabola.] 

Ex.  20.   Prove  that  the  area  of  a  closed  curve  is  represented  by 

iJ(^|-2/f)f?«[oriJ(x#-2,cZx)] 

taken  round  the  curve.      (See  Williamson,  Integral  Calculus,  Art,   139 ; 
Gibson,  Calculus,  §  128.) 


112.] 


VOLUMES   OF  REVOLUTION, 


199 


Fig.  55 


112.  Volumes  of  solids  of  revolution.  Suppose  that  the  arc  PQ 
of  the  curve 

revolves  about  the  a;-axis.  It  is  required  to  find  the  volume 
enclosed  by  the  surface  generated  by  PQ  in  its  revolution  and 
the  circular  ends  generated  by  the 
ordinates  AP  and  BQ.  (This  is  put 
briefly :  the  volume  generated  by  PQ.) 
Let  OA  =  a  and  OB  =  b. 

Suppose  that  AB  is  divided  into 
any  number  of  parts,  say  n,  each  equal 
to  Ax.  On  any  one  of  these  parts,  say 
LR,  construct  an  "inner"  and  an 
'^ outer"  rectangle,  as  shown  in  Fig.  55. 
Let  G  be  the  point  (x,  y),  and  K  be 
the  point  {x  -f-  A.t,  y  -f-  Ay).  When 
PQ  revolves  about  the  a^axis,  the  inner  rectangle  GR  describes  a 
cylinder  of  radius  GL  {i.e.  y),  and  thickness  Ax.  At  the  same 
time  the  outer  rectangle  KL  describes  a  cylinder  of  radius  KR 
(i.e.  y-{-Ay),  and  thickness  Ax.  It  is  evident  that  the  volume 
PQST  is  greater  than  the  sum  of  the  cylinders  described  by  the 
inner  rectangles,  and  is  less  than  the  sum  of  the  cylinders  described 
by  the  outer  rectangles.     That  is, 

sum  of  outer  cylinders  >  vol.  PQST  >  sum  of  inner  cylinders. 

The  difference  between  the  volume  of  the  outer  cylinders  and 
the  volume  of  the  inner  cylinders  approaches  zero  when  Ax 
approaches  zero.     Hence, 

vol.  PQST=  liniAx-o  J  sum  of  inner  (or  outer)  cylinders  J. 
That  is, 

voL  J*Q-S»T=  limAxio  [sum  of  cylinders  like  that  generated 
by  GR  when  x  increases  from  a  to  b\ 


limAx-o  /    (ttLG^  '  Ax)  =  IT  I         y^dx, 

*=«  (See  Art.  96.) 


200 


INFINITESIMAL   CALCULUS. 


[Ch.  XII. 


The  infinitesimal  differential  iri/dx, 
which  is  the  volume  of  an  infinitesimal 
cylinder  of  radius  y  and  infinitesimal  thick- 
ness dx,  is  called  an  element  of  voluyne. 

When  PQ  revolves  about  the  ?/-axis  the 
element  of  volume  is  evidently  irx^dy.  If 
the  ordinates  of  P  and  Q  are  c  and  d  respec- 
tively, the  volume  generated, 

vol.  PQTV=  Tc  (^^^oc^dy. 

Note  1.  It  is  almost  self-evident  that  the  volume  of  the  inner  cylinders 
and  the  volume  of  the  outer  cylinders  (Fig.  55),  approach  equality  -wtien 
their  thickness  Ax  approaches  zero. 

Note  2.     See  Art.  67  (e). 

EXAMPLES. 

1.    Find  the  volume  generated  by  the  revolution,  about  the  x-axis,  of  the 
part  of  the  line  3  x  +  10  j/  =  30  intercepted  between 
the  axes. 

The  given  line  is  AB.  The  element  of  volume 
is  -KXp-  dx.  At  B,  X  =  0  ;  at  ^,  x  =  10.  Accord- 
ingly, the  end-values  of  x  are  0  and  10.     Hence, 


rx=io  r: 

vol.  cone  ABC  =  tr  \        y^dx  =  Tr  \ 

Jx=0  Jo 


iV30-3x\2 


Fig.  57. 


V      10 
=  94.248  cubic  inches. 

2.  Verify  the  result  in  Ex.  1  by  finding  the  volume  of  the  cone  in  the 
ordinary  way. 

3.  Derive  by  the  calculus  the  ordinary  formula  for  finding  the  volume  of 
a  right  circular  cone  having  height  h  and  base  of  radius  a.     (See  Ex.  8.) 

4.  (a)  Find   the  volume    generated    by  the    revolution    of    the   ellipse 

9  x2  -f  16  ?/2  =  144  about  the  x-axis.  (b) 
Find  the  volume  bounded  by  a  zone  of  the 
surface  and  the  planes  for  which  x  =  2  and 
x  =  3. 

The  element  of  volume  is  -rry'^  dx. 
(a)  Vol.  ellipsoid 

=  2\oLABB'  =  2Tr  ('^^^dx 

Jx=0 

-  2z  r\l44  -  9  x2)dx  =  48  TT 
=  150.8  cubic  units. 


112.]  EXAMPLES.  201 

Or,  vol.  ellipsoid  =  ir  \        y^dx  =  150.8  cubic  units. 

(6)  Vol.  segment  PQQ'F'  =  ir  ("^y^dx  =  ^  ir  =  17.08  cubic  units. 

5.    Find  the  volume  generated  hy  revolving  the  arc  of  the  curve  y  =  x^ 
between  the  points  (0,  0)  and  (2,  8),  about  the  y-axis. 

The  arc  is  OA.     The  element  of  volume,  taking  any 
point  F(x,  y)  on  OA,  is  ttx-  dy.     Hence, 

vol.  OAB  =  IT  P"%2  dy  =  Tr  Cy^  dy  =  -%^-  ir 

Jj/=0  Jo 

=  60.32  cubic  units. 

The  integral  may  also  be  expressed  in  terms  of  x. 

Thus,  ^^^2 

vol.  OAB  =  ir\    ^x^dy. 


Since 


y  =  x^,  dy  =  ^  x^  dx. 


.'.  vol.  OAB  =  3  IT  Cx^  dx  =  9/  TT  =  60.32,  as  above. 


6.   Find  the  volume  generated  by  revolving  about  the  t/-axis  the  arc  of 


the  catenary 


y  =  ^(ea  +  e   a) 


between  the  lines  x  =  a  and  x  =  —  a.  AC  A'  is  the  catenary  ;  A  and  A'  are 
the  points  whose  abscissas  are  a  and  —  a  respec- 
tively. The  volume  generated  by  revolving  AC  A' 
about  OY  is  evidently  the  same  as  the  volume  gener- 
ated by  revolving  CA.  The  element  of  volume  is 
Tx2  dy.  *  ^^^ 

.-.  vol.  ACA<G  =  Tr\       .x2  dy.  (1 ) 

Jx=0 

In  this  case  it  is  easier  to  express  the  differential 
and  the  end-values  in  terms  of  x  than  in  terms  of 
y.     From  the  equation  of  the  curve  it  follows  that 


dy  =  \  (e'a  —  e  a)  dx. 
Hence  (1)  becomes  vol.  ACA'G  =-  \    (x^ eo  —  x'^ e  «)  dx. 

Integration  (by  parts)  of  the  terms  in  (2)  gives 

vol.  ACA'G  =  '^(e  +  ^-4:\=  .878  «». 


(2) 


202 


INFINITESIMAL   CALCUL US. 


[Ch.  XII. 


B 

T 

V 

i 

R 

\n* 

yi. 

M 

f 

4 

'\ 

0 

r     I 

X 

i 

r 

\ 

b" 

y 

S 

7.    Find,  by  the  calculus,  the  volume  of  the  ring  generated  by  revolv- 
ing a  circle  of  radius  5  inches  about  a  line  distant  7  inches  from  the  centre  of 
the  circle. 

Let  C  be  the  circle  and  ST  the  line.  Choose 
for  the  X-axis  the  line  passing  through  the  centre 
O  at  right  angles  to  ST,  and  take  OY  for  the 
y-axis.    Then 

the  equation  of  the  circle  is        x^  +  y^  =  25, 

and  the  equation  of  the  line  is  x  =  7. 

Through  any  point  P(x,  y)  on  the  circle,  draw 
YiG,  61.  P'PM  parallel  to  the  x-axis.     Suppose  that  PG^ 

at  right  angles  to  PP',  is  of  infinitesimal  length 
dy.  Then  the  rectangle  P'6r,  on  revolving  about  ST,  generates  an  infini- 
tesimal part  of  the  volume  of  the  ring.  The  limit  of  the  sum  of  these  parts 
as  y  changes  from  B'  to  B,  is  the  volume  required. 

The  volume  generated  by  P'G  =  ir  (PM^  -  PM^)  dy. 

Now  PM=1  -PB  =1  -  V25  -  y^, 

and  P>M=  7  +  BP'  =  1  -\-  \/25  -  y'\ 

.'.  vol.  generated  by  P'G  =  28  7r\/25  —  y^  •  dy. 


vol.  of  ring  =  2  ^"^28  TrV2b-y^dy=S50  ir^  cubic  units. 


[Or, 
as  in  Ex.  4  (a).] 


vol.  of  ring  =  \     .  28  v  V25  —  y'^  dy = 350  tr^  cubic  units, 

Jy=-5 


8.  Find  the  volume  of  a  cone  in  which  the  base  is  any  plane  figure  of 
area  A,  and  the  perpendicular  from  the  vertex  to  the  base  is  h. 

9.  Find  the  volume   generated  by  revolving  the   arc  BEC  (Fig.  53) 
about  the  x-axis. 

10.  Find  the  volume  generated  by  the  revolution  of  MTKN  (Fig.  52) 
about  the  x-axis. 

11.  Find  the  volume  generated  by  the  revolution  of  ORLM  (Fig.   51) 
about  the  y-axis. 

12.  Find  the  volume  generated  by  the  revolution  of  ABLB  (Fig.  50)  : 
(a)  about  the  ?/-axis  ;  (&)  about  the  x-axis. 

13.  Find  the  volume  generated  by  revolving  the  loop  in  Ex.  18,  Art. 
Ill,  about  the  x-axis. 


113.]  EQUATIONS  OF  CURVES.  203 

14.  Find,  by  the  calculus,  the  volume  generated  by  the  revolution  about 
the  X-axis,  of  the  part  of  each  of  the  following  lines  that  is  intercepted  bertween 
the  axes,  and  verify  the  results  by  the  ordinary  rule  for  finding  the  volume 
of  a  cone  : 

(1)  3x  +  iy  =  2;  (3)  7a:  +  32/ +  20  =  0; 

(2)2x-5y  =  7',  (^i)  Sx  -  4y  +  10  =  0. 

.  15.  Find  the  volume  generated  by  the  revolution  about  the  y-axis,  of 
each  of  the  intercepts  in  Ex.  14,  and  verify  the  result  by  the  usual  method 
of  computation. 

16.  Find  the  volume  generated  when  each  of  the  figures  described  in 
Ex.  17,  (l)-(9),  Art.  Ill,  revolves  about  the  ar-axis. 

17.  Find  the  volume  generated  when  each  of  the  figures  in  Ex.  16 
revolves  about  the  y-axis. 

18.  The  figures  bounded  by  a  quadrant  of  an  ellipse  of  semi-axes  9 
and  5  inches  and  the  tangents  at  its  extremities  revolves  about  each  tangent 
in  turn :  find  the  volumes  of  each  of  the  solids  thus  generated. 

19.  Find  the  volume  of  a  sphere  of  radius  a,  considering  the  sphere 
as  generated  by  the  revolution  of  a  circle  about  one  of  its  diameters. 

Note  3.  The  volume  of  a  sphere  may  also  be  obtained  by  considering  the 
sphere  as  made  up  of  concentric  spherical  shells  of  infinitesimal  thickness. 
The  volume  of  a  shell  whose  inner  radius  is  r  and  whose  thickness  is  an  infini- 
tesimal dr  is  (to  within  an  infinitesimal  of  lower  order)  4  wr'^  dr.  Accordingly, 
volume  of  sphere  =  I   4  wr'^  dr  =  ^  ira^. 

20.  Find  the  volume  generated  by  the  revolution  of  the  hypocycloid 
x^+  y's  =  as  about  the  x-axis.     {Ans.  ^^^  ira^.) 

113.  Derivation  of  the  equations  of  curves.  The  equation  of  a 
curve  or  family  of  curves  can  be  found  when  a  geometrical  prop- 
erty of  a  curve  is  known.  Exercises  of  this  kind  constitute  an 
important  part  of  analytic  geometry.  For  instance,  the  equation 
of  a  circle  can  be  derived  from  the  property  that  the  points  on 
the  circle  are  at  a  given  common  distance  from  a  fixed  point. 
The  statement  of  a  geometrical  property  possessed  by  a  curve 
may  involve  derivatives  or  differentials.  To  derive  the  equation 
of  the  curve  from  this  statement  is,  quite  frequently,  a  difficult 
problem.  There  are  a  few  simple  cases,  however,  in  which  it  is 
possible  to  find  the  equation  of  the  curve  by  means  of  a  knowl- 
edge of  the  preceding  articles.  A  few  very  simple  examples 
have  been  given  in  Art.  97. 


204  INFINITESIMAL   CALCULUS.  [Ch.  XII. 

Note  1 .  It  may  be  worth  while  merely  to  glance  at  more  difficult  problems 
of  this  kind  and  at  the  text  relating  thereto,  in  Chapter  XXI.  and  in  Murray's 
Introductory  Course  in  Differential  Equations^  Chaps.  V.  and  X.  Also  see 
Cajori,  History  of  Mathematics^  ^^^.201 -20'^^  "  Much  greater  than  .  .  .  integral 
of  it." 

Note  2.  It  has  been  shown  in  Arts.  58,  59,  that  for  the  curve  whose 
equation  is  f(x,  y)  =  0,  rectangular  coordinates,  if  (x,  y)  denotes  any  point 
on  the  curve  and  m  is  the  slope  of  the  tangent  at  (x,  ?/),  then 

m  =  -^ ;    subtangent  =  y—;    subnormal  =  y ^• 
dx  dy  dx 

Note  3.  It  has  been  shown  in  Arts.  60,  61,  that  for  the  curve  whose 
equation  is  /(r,  6)  =  0,  if  (r,  6)  denotes  any  point  on  the  curve,  xp  the  angle 
between  the  radius  vector  and  the  tangent  at  this  point,  and  <p  the  angle 
which  the  tangent  makes  with  the  initial  line,  then 

tan\^  =  r— ;  0  =  1//  +  ^; 
dr 

do  dr 

polar  subtangent  =  y2  —  ;    polar  subnormal  =  — 


N.B.     Draw  the  curves  in  the  following  examples. 


EXAMPLES. 

1.    A  curve  has  a  constant  subnormal  4  and  passes  through  the  point 
(3,  5)  :  what  is  its  equation  ? 

Here  the  subnormal,  y-^  =  i. 

dx 

On  using  differentials,  ydy  =  4:  dx. 

Integration  gives  ^  +  ci  =  4  x  +  C2  ; 

whence  ^  =  4x  -\-  k,  in  which  k  =  C2  —  Ci. 

Since  (3,  5)  is  on  the  curve,    "^f  =  12  +  A:,  whence  k  —  \. 

?/2  1 

Accordingly,  ^  =  4  x  +  - ,  <*.e.  y'^  =  8  x  +  1,  is  the  equation. 

Note  4.  In  working  these  examples  it  is  enlivening 
and  helpful,  to  express  the  given  conditions  by  means 
of  a  figure.  This  tentative  figure  ca;n  be  corrected 
when  fuller  information  is  derived.  Thus,  for  Ex.  1 
draw  a  curve  passing  through  (3,  5),  and  at  any  point 

-4 \N         P(x,  y)    on   this    curve    make   the   construction  in 

Fig.  62.  Fig.  62   showing  the  subnormal  4.      Here  Z.  MPN 

A  HPT.     Now  tan  JfPiV  =  -,  i.e.  ^  =  -.     Then  proceed  as  above. 
y         dx     y 


113.]  EXAMPLES.  205 

2.  A  curve  has  a  constant  subnormal  and  passes  through  the  points 
(2,  4),  (3,  8)  :  find  its  equation  and  the  length  of  the  constant  subnormal. 

3.  A  curve  has  a  constant  subtangent  2,  and  passes  through  the  point 
(4,  1)  :  find  its  equation. 

4.  Determine  the  curve  which  has  a  constant  subtangent  and  passes 
through  the  points  (4,  1),  (8,  e)  :  find  its  equation  and  the  length  of  the 
subtangent. 

5.  Find  the  curve  in  which  the  length  of  the  subtangent  for  any  point 
is  twice  the  length  of  the  abscissa,  and  which  passes  through  (3,  4). 

6.  In  what  curves  does  the  subnormal  vary  as  the  abscissa  ?  Deter- 
mine the  curve  in  which  the  length  of  the  subnormal  for  any  point  is  pro- 
portional to  the  length  of  the  abscissa,  and  which  passes  through  the  points 
(2,  4),  (3,  8). 

7.  In  what  curves  does  the  slope  vary  as  the  abscissa  ?  Determine 
the  curve  in  which  the  slope  at  any  point  is  proportional  to  the  length  of  the 
abscissa,  and  which  passes  through  the  points  (0,  2),  (3,  5). 

8.  In  what  curves  does  the  slope  vary  inversely  as  the  ordinate  ? 
Determine  the  curve  in  which  the  slope  at  any  point  is  inversely  proportional 
to  the  length  of  the  ordinate  and  which  passes  through  the  points  named  in 
Ex.  7. 

9.  Determine  the  polar  curves  in  which  the  tangent  at  any  point 
makes  with  the  initial  line  an  angle  equal  to  twice  the  vectorial  angle.     Which 

of  these  curves  passes  through  the  point  (4,  -  ]  ? 

10.  Determine  the  polar  curves  in  which  the  subtangent  is  twice  the 
radius  vector.    Which  of  these  curves  passes  through  the  point  (2,  0")  ? 

11.  Determine  the  polar  curves  in  which  the  subnormal  varies  as  the  sine 
of  the  vectorial  angle,  and  which  pass  through  the  pole. 


CHAPTER    XIII. 

INTEGRATION  OF  IRRATIONAL  AND  TRIGONOMETRIC 

FUNCTIONS. 

114.  The  integration  of  differential  expressions  involving  irra- 
tional quantities  and  trigonometric  quantities  will  now  be  con- 
sidered. Examples  of  this  kind  and  methods  of  treating  them 
have  already  been  given  in  preceding  articles.  (See  Art.  104, 
Art.  105,  Exs.  10-18.)  Only  a  few  very  special  forms  are  dis- 
cussed in  this  book. 

Note.  Chapter  XI.  provides  a  good  part  of  the  knowledge  of  formal  inte- 
gration sufficient  for  elementary  work  in  physics  and  mechanics  and  for  the 
ordinary  problems  in  engineering.  Accordingly,  this  chapter  may  be  merely 
glanced  at  by  those  who  have  only  a  very  short  time  to  give  to  the  study  of 
the  calculus  and  thus  find  it  necessary  to  take  on  faith  the  results  given  in 
tables  of  integrals. 

INTEGRATION  OF  IRRATIONAL   FUNCTIONS. 

115.  The  reciprocal  substitution.  This  substitution,  which  some- 
times leads  to  an  easily  integrable  form,  has  been  shown  in  Art. 
107,  Ex  6.     Additional  exercises  are  here  appended. 

Ex.  1.    Find  ( ^ 


Put  x  =  -'     Then  dx  =  -^dt;   and 
t  t^ 

r dx ^  ^ _  r      tdt       ^  J-  ("(i  _  a2f2)-J^(i  _  aH^). 

Exs.  2-9.   Derive  integrals  23,  26,  27,  39,  42,  43,  54  a,  59  a,  61  a,  pages 
403-406. 

2(16 


lU-116.]  IRRATIONAL  FUNCTIONS.  207 

Note.  Trigonometric  substitutions.  Examples  of  a  useful  trigonometric 
substitution  have  been  given  in  Art.  105,  Exs.  4,  5.  A  differential  expression 
in  which  Va-  +  x'^  occurs  may  sometimes  be  simplified  for  purposes  of  inte- 
gration by  substituting  a  tan  6  for  x,  and  expressions  containing  Vx^  —  a^ 
by  substituting  a  sec  d  for  x. 

For  instance,  in  Ex.  1  put  x  =  a  sec  6.  Then  dx  =  a  sec  6  tan  6  dd  ;  and 
dx         _  1   r_„  .  ,,.  _  1  g.^^  ^  Va;2  _  a-2 


r        ^^-         ^Ifcos^c^^ 


i-i  a^x 


116.  Differential  expressions  involving  -\/ff  +  6jr.  By  this  is 
meant  differentials  in  which  the  irrational  terms  or  factors  are 
fractional  powers  of  a  single  form,  a  -f  hx.  (In  particular  cases  a 
may  be  0  and  h  may  be  1 ;  the  irrational  terms  or  factors  are  then 
fractional  powers  of  x.)  For  preceding  instances  see  Art.  105, 
Ex.  3,  and  Exs.  4,  10  at  the  end  of  Chapter  XI. 

If  n  is  the  least  common  denominator  of  the  fractional  indices 
of  a  +  hx,  the  expression  reduces  to  the  form 


F{x,  V  a  -t-  hx)  dx.  (1) 

This  can  be  rationalised  by  putting 
a-\-hx  =  z". 

For  then  x  =  — ^—  and  dx  =  -z''~^dz\    and,  accordingly,  ex- 


pression (1)  becomes 


h     \     h 


This  is  rational  in  z,  and  accordingly  may  be  integrated  by  the 
preceding  articles. 


Ex 


Y    Cx_d^^  Ex.4.  r(3  +  a:)\/(2  +  x)^^a;. 
*^  1  +  X*  -^ 

.  2.  Cy^^.  Ex.  5.  r  ^^ 

J  x  +  1  J 


y/2  -  x(7  -I-  5  V2  -  x) 


Ex.  3.   f        ^^^        .  Ex.  6.   r ^^^+i  dx. 

''v/(3x-2)4  ^Vx  +  1-1 


208  INFINITESIMAL   CALCULUS.  [Ch.  XIII. 


117  A,  Expressions  of  the  form  f  (jr,  VjP  +  a^r  +  6)  (/jr.  B.  Ex- 
pressions of  the  form  F(jr,  -^  —  x'^  -{- ax  +  b)dx  )  F{u^  v)  being  a 
rational  integral  function  of  u  and  v. 

A,   The  first  expression  can  be  rationalised  by  putting 


Va^  -\-  ax  -\-h  =  z  —  X,  (1) 

and  changing  the  variable  from  a;  to  2. 

For,  on  squaring  and  solving  Equation  (1)  for  x, 


From  this, 


a-\-2z 

(2) 

(3) 

On  substituting  the  value  of  x  in  (2)  in  the  second  member 
of(l).  , 

a  +  2  2; 
Accordingly, 

Fix,  ^x^^ax+h)dx  becomes  2 f(^,  t+^Vl+^dz. 


This  is  rational  in  z,  and,  accordingly,  may  be  integrated  by 
preceding  articles. 


Ex.  1.   Find  (        ^^^ 

•^  Vx^-x  +  l 


Assume 

Vx^- 

-x-\-l  =  z-x. 

From  this, 

2«-l 

Then 

(2^-1)^    "• 

and  Vx'^-x  +  l=z-x=  ?!^U?-±i. 

2z  —  1 


117.]  IRRATIONAL   FUNCTIONS.  209 

On  substitution  of  these  values  in  the  given  integral, 

r      xdx      ^2(-^'^^^dz  =  hz  + 5 ^  +  iogV27::^  +  c 

J  Vx2-x  +  l        ^C^^-l)-'  4(2^-1) 

^  (See  Art.  108.) 

_  a;  +  Voc^  -  X  +  1      3 

2  4  (2  X  -  1  +  2  Vx2  -  X  +  1) 


+  ^  log  (2  X  -  1  +  2  Vx^  -  X  +  1)  +  c 


=  J  log  (2  X  -  1  +  2  Vx^  _  X  +  1)  +  Vx'^  -  X  +  1  +  ^. 

(A:  =  i  +  c.) 

It  happens  that  this  is  not  the  shortest  way  of  working  this  particular 
example ;  but  the  above  serves  to  show  the  substitution  described  in  this 
article.  The  integral  may  also  be  obtained  in  the  following  way ;  this 
method  is  applicable  to  many  integrals. 

r      xdx      ^r/i.     2X-1     ^1 1 N^^ 

•^  Vx2  -  X  +  1     -^  V2     Vx2-x  +  l      2     Vx^  -  X  +  1  / 
=  1 J  (x2  -  X  +  l)-*d(x2  _  a;  +  1)  +  J I 


dx 


V(x  -1)2  +  1 


Vx2  -  X  +  1  +  I  log  (X  -  i  +  Vx2  -  X  +  1)  +  C 


=  \/x2  -  X  +  1  +  J  log  (2  X  -  1  +  2  \/x2  -  X  +  1)  +  Ci. 
x-3  2 


Ex.2,  r  ^^-^)^^  =u    ^-'^     -      ^      ^ 

*^Vx2-6x  +  25     *'Wx2-6x-f25     VCx  -  3)2  +  16/ 


dx 


=  Vx2  -  6x  +  25  -  2  log  (x  -  3  +  Vx2  -  6  X  +  25). 

B.    Suppose  that  -  x^  +  ax -^b  =  (x- p)(q  -  x). 

The  second  expression  at  the  head  of  this  article  can  be  rational- 
ised by  putting 


V— x^  -f-aa^H-  6,  i.e.  V(x  —  p)(q  —  x)  =  (x  —  p) z,  (3) 

and  changing  the  variable  from  x  to  z. 

On  squaring  in  (3),.  q  —  x=(x—p)z^; 

on  solving  for  x,  ^  —  h ^  5  W 

2  zip q) 

whence,  on  differentiation,  dx  =  -—~ — ^  dz. 

(1  -f  zy 


210  INFINITESIMAL   CALCULUS.  [Ch.  XIII. 


Substitution  of  the  value  of  x  in  (4)  in  the  second  member  of 

),  gives 

Accordingly, 


1+5;- 


'pz^-{-q  {q—p)z\    zdz 


F(x,  V-x'+ax+b)dx  becomes  2  (p-q)F  i'-^-^,  ^^^—^^^  \-J^^—, 

\l+z'      l-\-z'  ){l+zy 

This  is  rational  in  z,  and,  accordingly,  may  be  integrated  by 
preceding  articles. 

Note  1.     Instead  of  (3)  the  relation  ^ 


\/{x  —  p)  {q  —  x)  —  {q  —  x)z 
may  be  used. 


Note  2.     If   v  ±;«-  -^  qx  -\-  r  occurs,  it  may  be  reduced  to  form  A  or 

B  :  thus,   Vp  a/  ±  a--  +  -a;  +  -  • 


P  P 

EXAMPLES. 


3.    Find    f -J^ 


X  \/l2  —  X  —  X?- 


Put  V12  -  X  -  x2  =  -yJix  +  4)  (3  -  x)  =  (x  +  4)0. 

From  this,  on  squaring,  3  —  x  =  (x  +  4)2;2. 

3  _  4  ^2 

On  solving  for  x,  x  = — -• 

1  +  s'^ 

Accordingly,  dx  =  ~  ^^  ^f  ^,  \/l2  -  x  -  x'-^  =  (x  +  4)  -  -    ^^ 


*^  X  V'12  -  X  -  x'^         J  40'2 - 3     2  V3 


(l+0'-^)2'  l+^I--^ 

2  0  -  V3 


2  0  +  V3~ 


log 


2V3-x-V3(x  +  4). 


2\/3       2V3^+V3(x+4) 

4.    Solve  Ex.  3,  using  the  substitution  Vl2  -  x  -  x^  =  (3  -  x)  «. 

5      r      (2  X  +  5)  c?x  g_     r     (3x-4)t?x 

J  \/4  x2  +  6  X  +  11  -^  Vl2  -  4  X  -  x2 


X  V12  -  4  X  -  x'-^ 
8     r     (3  a;  -  4)  dx  r3  x  -  4  ^  3     4  n 

•'xVl2-4x-x-^  L     X  a;  J 


118.]  IRRATIONAL  FUNCTIONS.  211 

(Put  x^2  =  z.) 


"h 


x  Vx2  +  a;  +  1  -^    X  Vx2  +  x  +  1 


X  4-  2)  Vx2  +  4  X  -  12 

2m+l 

Note  3.     The  integi^ands  in  integrals  of  the  form  \  xP(a  4-  bx^)    ^   dx  in 

which  iu  is  any  integer  and  p  is  an  odd  integer,  positive  or  negative,  can 
be  rationalised  by  means  of  the  substitution  a  +  bx^  =  z^.     Thus : 

12.  C^^^d^. 
^  Vx2  -  a^ 

Put  x2-a2  =  22. 

Then  xdx  =  zdz  ; 

and       f     ^^^^^^     :=  f  (22  +  a2)d^=?(;s2  +  3(^2)  ^  y'^  +  2  q^  ^^._,  _  ^^ 
J  Vx2  -  a^     -^  3  3 

13.  Find    f ^^       ■    (see  Formula  XXI.,  Art.  107):    (1)    Using  the 

•^  x  \/x2  —  d^ 

substitution  x  =  a  sec  6  ;    (2)   using  the  substitution  x  =  -  \    (3)    using  the 

t 
substitution  x^  —  a2  —  ^2_      (Show  the  equivalence  of  the  various  forms  of 
the  integral.) 

14.  Show  the  truth  of  the  statement  in  Note  3. 

118.   To  find  I  jr'"(a  +  bx^ydx.    Here  m,  n,  and  p  are  constants, 

positive  or  negative,  integral  or  fractional.  The  given  integral, 
as  will  be  shown  in  the  working  of  examples,  can  be  connected 
with  simpler  integrals  in  a  particular  way.  By  "a  simpler  inte- 
gral" is  meant  one  that  is  simpler  from  the  point  of  view  of 
integration.  For  instance,  if  m  =  5,  the  integral  in  which  m  =  3, 
other  things  being  the  same,  is  simpler ;  ii  p  =  —  ^,  the  integral 
in  which  p  =  —  i  other  things  being  the  same,  is  simpler.  It  will 
be  found  that  tJie  given  integral  can  be  connected  with  an  integral  in 
which  the  m  is  increased  or  decreased  by  n,  or  icith  an  integral  in 
which  the  p  is  increased  or  decreased  by  1 ;  i.e.,  with  one  or  other 
of  the  four  inteo^rals  : 


r^,«+n(^^  +  ba^ydx,  Cx"'(a  +  bxy+'  dx 

Cx'^-''(a  +  bx'^y  dx,  Cx"Xa  +  bx'^y "  ^  dx.  ^ 


(a) 


212  INFINITESIMAL   CALCULUS.  [Ch.  XIII. 

When  one  of  these  four  integrals  is  chosen,  a  relation  between 
it  and  the  required  integral  can  be  expressed  in  the  following 
way: 

Form  a  function  of  x  in  which  the  x  oiitside  the  bracket  has  an 
index  one  greater  than  the  least  index  of  the  corresponding  x  in  the 
required  and  the  chosen  integrals,  and  in  which  the  bracket  has  an 
index  one  greater  than  the  least  index  of  the  bracket  in  those  integrals. 
Give  the  function  thus  formed  an  arbitrary  constant  coefficient  and 
give  the  chosen  integral  an  arbitrary  constant  coefficient;  equate  the 
sum  of  these  quantities  to  the  required  integral.  The  value  of  the 
arbitrary  coefficients  can  then  be  determined. 

For  example,  let 

j  x^{a -\- bx'^y dx  be  connected  with    I  xf^{a-\-bx''y-'^dx. 

The  function  formed  by  the  rule  is  x'^^^{a  -f  bx'^y.     Put 
Cx'^ia  +  bx'^ydx  =  Ax'^+Xa  +  bx^  4-  B  C<c'^(a  +  bx'^y-^  dx,     (1) 

in  which  ^  and  5  are  arbitrary  constants. 

It  is  now  necessary  to  find  such  values  for  A  and  B  as  will 
make  (1)  an  identical  equation. 

In  order  to  determine  A  and  B,  take  the  derivatives  of  both 
members  of  (1),  simplify,  and  then  equate  coefficients  of  like 
powers  of  x.     Thus,  on  differentiating  the  members  of  (1), 

x'^ia  -f  bx'^y  =  A(m  -\-  l)x'^(a  +  bx'^y  +  Ax'^+^pia  +  bx'^y-hibx''-'^ 

+  Bx'^(a  +  bx'^y-K 

On  division  by  x^(a  -f  bx^y~'^,  and  simplification, 

a  -|-  6cc"  =  Ab(m  +  np-\-l)x'^  -\-  Aa(m  +  1)  -j-  R 

On  equating  coefficients  of  like  powers  of  x  and  solving  for  A 
and  B, 

A  = i -,       B=       ""^       • 

m  +  nj)  +  1  m-\-np-{-l 


118.]  IBRATIONAL   FUNCTIONS.  213 

The  substitution  of  these  values  in  (1)  gives 

J        ^  ^  m  +  np  +  1 

attp —    Cucnif^a  +  6ic»*)P-i  dx. 

m  +  np  -\-  IJ  ^ 

On  connecting  the  required  integral  with  each  of  the  other 
integrals  in  (a)  and  proceeding  in  a  similar  manner,  the  results 
(1),  (2),  (4),  page  401,  are  obtained.  The  deduction  of  them  is 
left  as  an  exercise  for  the  student. 

Note  1.  Formulas  1^,  page  401,  are  examples  of  what  are  usually  termed 
Formulas  of  Reduction.  Frequently  integrals  are  obtained  by  substituting 
the  particular  values  of  m,  w,  p  in  these  formulas  of  reduction.  To  memorize 
such  formulas  is,  however,  a  waste  of  energy  ;  it  is  better,  at  least  for 
beginners,  to  integrate  hy  the  method  whereby  these  formulas  have  been 
obtained. 

Note  2.  It  will  be  observed  that  some  of  these  formulas  fail  for  certain 
values  of  m,  n,  p ;  viz.,  when  w  +  wp  +  1  =  0,  when  ?/i  =  —  1,  and  when 
p  =  —\.  Other  formulas  or  other  methods  may  be  applied  in  each  of  these 
cases. 

Note  3.  Its  success  may  be  regarded  as  one  proof  of  the  above  method. 
In  the  large  majority  of  text-books  on  calculus,  formulas  1,  2,  3,  4,  page  401, 
are  derived  in  a  straightforward  way  by  integration  by  parts.  For  this 
derivation  see  almost  any  calculus,  e.g.  Murray,  Integral  Calc2ilus,  Appendix, 

Note  B.     For  other  formulas  of  reduction  for  ix'"(a  +  bx^)Pdx.,  obtained 

by  the  method  of  "connection"  or  "arbitrary  coefficients,"  see  Edwards, 
•Integral  Calculus,  Art.  82,  and  integrals  5,  6,  page  402. 

EXAMPLES. 

1.   Find  f ^^  i.e.    ( x-^  (x^  -  a'^)'^ dx.       (See  Ex.  1,  Art.  115.) 

-^  xWx^  -  a2  J 

Here  w  =  —  2,  n  =  2,  p  =  —  |.     The  best  integral  to  connect  with  is 

.^—.     On 


obviously  the  integral  in  which  the  m  is  raised  by  2,  viz.  i  - 

•^  Vx^  -  a2 
making  the  connection  according  to  the  directions  given  above, 

(1)  f  x-2  (a;2  -  a^)~^  dx  =  Ax-^{x^  -  a^)^  -^b({x^-  a"^)'^ dx. 

It  is  now  necessary  to  find  such  values  for  A  and  B  as  will  make  this 
equation  an  identical  equation. 


214  INFINITESIMAL   CALCULUS.  [Ch.  XIII. 

On  differentiation,  and  equating  the  derivatives, 

x-2  (a;2  -  «2)-?  :=  _  Ax-^  (x^  -  a2)^  +  ^  (a;2  -  a^)~^  -\-  B(x'^-  a^yK 
On  simplifying,  by  multiplying  through  by  x^(x'^  —  a^y, 
1  =  -  ^  (x2  -  a2)  _^  Ax2  4-  J5x2. 

On  equating  coeflBcients  of  like  powers  of  x, 

B  =  0  and  Aa^  =  1  :   whence  A=—. 
On  substitution  of  these  values  of  A  and'  ^  in  (1), 


J; 


dx  \/x2  —  a''^ 


xWx^  -  a2  a^x 

2.    Find    C-J^^^L—,  i.e.     ( x^  (x^- -  a^yi dx.     (See  Ex.  12,  Art.  117.) 

'^  Vx2  -  a2  J 

Here  m  =  3,  w  =  2,  j?  =  —  ^.  It  will  obviously  be  an  advantage  to  lessen 
m.  Accordingly,  let  connection  be  made  with  (  x(x2  —  d^y^  dx.  On  doing 
this  in  the  way  described, 

(1)  f  x3  (x2  -  a2)-?  ax  =  Ax^  (a;2  -  ^2) *  +  j^  C^  (^,2  _  f,2y2  ^x. 

It  is  now  necessary  to  find  such  values  for  A  and  B  as  will  make  this  an 
identical  equation. 

On  taking  the  derivatives  and  equating  them, 

x^  (a;2  -  a2)-?  =  2  Ax  {x?  -  a'^y  +  Ax'^  {x?  -  aPy^  +  Bx  {x^  -  a^y'^. 

On  simplifying,  by  dividing  through  by  x{x'^  —  a^y^, 

x^  =  2A(x^-  oP-)  H-  Ax^  +  B. 

On  equating  coefficients  of  like  powers  of  x,  and  solving  for  A  and  5,  it  is 
found  that  A  =  \,  B  =  la^. 
Substitution  in  (1)  gives 

xdx 


f-^^j^ = i  x2  v^^^T^^ + f  a2  r 

J   a/'»-2  _  nl  ♦^ 


Vx2  -  a2  ^  Vx2  -  a^ 


=  ^  x2  V^^^^  +  I  aW^t^^^^  =  (x2  +  2a2)vV2-a2, 

o 

3.    Find  f ^^ ,  i.e.  ( (x"^ -\- a^y^  dx. 

J  (x2  +  a2)*'         J  ^     ^     ^ 

Here  m  =  0,  w  =  2,  j)  =  —  A:.    In  this  case  it  is  better  to  increase  p.     On 
proceeding  according  to  the  rule, 

(1)  f  (x2  +  a^y^dx  =  ^x(x2  +  rt2)-fc+i  +  7?  r(a:2  +  a^)-fc+idx. 


119.]  IRRATIONAL   FUNCTIONS.  215 

On  differentiation,  simplification  of  the  resulting  equation  by  division  by 
(x2  +  a^)-*,  equating  coefficients  of  like  powers,  solving  for  A  and  B,  and 
substitution  of  their  values  in  (1),  it  will  be  found  that 

C       dx        ^  1  i  __^___  +  C2  ^  -  3)  (         ^  ] 

4.  Derive  ( Va^  -  x:^  dx  by  this  method.     (See  Ex.  5,  Art.  105,  Ex.  5, 
Art.  106.) 

5.  Do  Ex.  16,  Art.  107,  by  this  method. 

Note  4.  It  is  sometimes  necessary  to  repeat  the  operation  of  reduction 
two  or  more  times. 

6.  Derive  integrals  21,  22,  23,  28,  30,  35,  40,  41,  42,  44,  pages  403-404, 
and  others  of  the  collection. 

7.  Derive   integrals  48,    53,    54,    55,  57,  pages  405-406  [  \/2  ax  ±  x-  = 
x^(2  a  ±  x)^].     (Compare  Exs.  6,  7,  and  Exs.  2-9,  Art.  115.) 

8.  Derive  formulas  1-6,  page  401. 

9.  Find        (^^^'^'dx^-^^'-^^y-, 

J        x"  3«-^x3 

(—^—-cU  =  -  {  3  a*sin-i^  -  2^(2  x^  +  3  a^)  Va^  -  x"-^ "I . 
J  Vrt2  -  x2  8  I  a  ) 

10.  Using  integrals  1-4  as  formulas  of  substitution  for  the  values  of  m,  7i, 
jf>,  «,  &,  derive  some  of  the  integrals  21-30,  37-46,  53-61,  pages  403-406. 

Note  5.  On  the  integration  of  irrational  expressions  also  see  Snyder  and 
Hutchinson,  Calculus,  Arts.  129-131,  139,  140.  These  articles  convey  valu- 
able additional  information,  and,  in  particular.  Art.  139  gives  an  interesting 
geometrical  interpretation  concerning  the  rationalisation  of  the  square  root 
of  a  quadratic  expression.     Also  see  the  references  given  in  Art.  122,  Note  2. 

INTEGRATION  OF   TRIGONOMETRIC   FUNCTIONS. 

N.B.  On  account  of  the  numerous  relations  between  the  trigonometric 
ratios,  the  indefinite  integral  of  a  trigonometric  differential  can  take  many 
forms. 

119.  Algebraic  transformations.  A  differential  expression  in- 
volving only  trigonometric  ratios  can  be  transformed  into  an 
algebraic  differential  by  substituting  a  variable,  t  say,  for  one  of 
the  trigonometric  ratios.  The  algebraic  differential  thus  obtained 
may  possibly  be  integrated  by  some  method  shown  in  the  preced- 
ing articles.     Knowledge  as  to  what  substitution  wiK  be  the  most 


216  INFINITESIMAL    CALCULUS,  [Ch.  XIII. 

convenient  one  to  make  in  a  given  case  can  best  be  acquired  by 
trial  and  experience.  Illustrations  of  this  article  have  already 
been  met  in  Art.  105,  Exs.  10,  11,  16,  17. 

Ex.  1.    See  exercises  just  referred  to. 

Ex.  2.    Do  Exs,  1-5,  7-9,  Art.  120,  making  algebraic  transformations. 

120.   Integrals  reducible  to   |  F(u)  du,  in  which  u  is  one  of  the 
trigonometric  ratios. 

(a)    I  sin**  a?  doc  and   |  cos**  oo  doc  are  thus  reducible  when  n  is  an 
odd  positive  integer.     For 

I  sin"  xdx=  I  sin**"^ x  •  sin  xdx  —  —  (  (1  —  cos^  x)~^  d  (cos  x). 

The  latter  form  can  be  expanded  in  a  finite  number  of  terms,     ~" 

being  an  integer,  and  then  integrated  term  by  term.  |  cos"  x  dx 
can  be  treated  similarly. 

EXAMPLES. 

1.  I  cos^x^^x  =  j  cos^x  •  cosicdcc  =  j  (1  —  sin2x)2d(sina;) 

=  1(1—2  sin2  X  +  sin*  x)  d  (sin  x)—  sin  x  —  |  sin^  x  -{■  \  sin^  x  -\-  c. 

2.  I  sin^  X  dx,    i  cos-^  x  dx,    \  sin^  x  dx. 

(6)     j  sin"ircos*^a5c?a?  is  thus  reducible  when  either  n  or  m  is  a 
positive  odd  integer. 

3.  I  sin^  xcos^xdx  =  \  sin2  x  cos 2  x  sin  x  dx 

=  —  I  (1  —  cos2  x)  cos2  X  d  (cos  aj)  =  —  I  (cos^  x  —  cos^  x)  d  (cos  x) 

1  n 

=  —  f  C0S2  X  4-  ^Y  COS  ^  X  +  c.  ^ 

4.  (1)    f^il^^dx,  (2)    (cos'^xsm^xdz,  (3)    f52££^, 

y  v^cosx  "^  -^   Vsinx 

(4)    j  cos^  X  sin^  x  t^x. 

Note.    Case  (a)  is  a  special  case  of  (6). 


120,  121.]  TRIGONOMETRIC  FUNCTIONS.  217 

(c)  I  sec**  a?  dx  and  I  cosec"  x  dx  are  thus  reducible  when  n  is 
a  positive  even  integer. 

5.  Tcosee^  xdx  =  \  cosec*  x  •  cosec^  xdx  =  —  i  (1  +  cot^  x)^  d  (cot  x) 

=  -  cot  X  (1  +  I  COt2  X  +  ^  cot*  X). 

6.  Show  the  truth  of  statement  (c). 

7.  (1)   I  sec*  X  dx,     (2)    j  cosec*  x  dx,     (3)    |  sec^  x  dx. 

(d)  I  taii»»*  a?  sec»*  ac  dx  and  i  cot"*  x  cosec»»  x  dx  are  thus  reduci- 
ble when  n  is  a  positive  even  integer^  or  when  m  is  a  positive  odd 
integer. 

8.  Show  the  truth  of  statement  {d). 

9.  (1)  i  tan2  X  sec*  X  (?x,  (2)  i  sec^  x  Vtan  x  (^x,  (3)  |  tan^  x  sec*  x  dx, 
(4)   jtan^  X  sec^  X  (?x,    (5)   j  cot^  x  Vcosec  x  <Zx,    (6)  cot^  x  cosec^  x  (?x. 

121.  Integration  aided  by  multiple  angles.  It  is  shown  in 
trigonometry  that 

sin  u  cos  w  =  Y  ^^^  ^  ^*> 

sin-  u  =  \(l  —  cos  2 u), 
cos^  u  =  i  (1  +  cos  2  u). 

Accordingly,  if  n  and  m  are  positive  even  integers,  sin**  x,  cos"  x, 
and  sin**  X  cos"*  .T  can  be  transformed  into  expressions  which  are 
rational  trigonometric  functions  of  2  x.  Differential  expressions 
involving  the  latter  are,  in  general,  more  easily  integrable  than 
the  original  differential  expressions  in  x. 

Ex.  1.    rcos*xdx=  ({\{\-\-co&2x)fdx  =  jf  (1  +  2  cos2  x  +  cos2  2x)  dx. 

Now       I  2  cos 2  xdx  =  sin 2 X,      and        j  cos2  2xdx  =  ^  |  (1  +  cos4x)  dx  = 

\{x  +  ^  sin  4  x).     .•.   \  cos*  x  dx  =  |  x  4-  5  sin  2  x  +  ^^  sin  4  x  +  c. 

Ex.  2.   \  sin2  X  cos^  x  dx  =  ^  |  sin^  2  x  dx  =  |^  j  (1  —  cos  4  x)  dx 

=  \x  —  j^^  sin 4  X  4-  c. 
Ex.3.    (1)    (sin*  xdx,  (2)    jcos^xdx,  (3)    j  sin*  x  cos*^  x  dx, 

(4)    j  sin^  X  cos^  X  dx,         (5)    |  sin*  x  cos*  x  dx. 


218  INFINITESIMAL   CALCULUS.  [Ch.  XIII. 

122.  Reduction  formulas.  There  are  several  formulas  which 
are  useful  in  integrating  trigonometric  differentials.  A  few  of 
them  are  deduced  here  ;  the  deduction  of  the  others  is  left  as  an 
exercise  for  the  student. 

(a)  To  find  A:  \  sin^'xdx,  and  B:  |  cos'^xdx,  when  n  is  any 
integer. 

A.   Integrate  by  parts,  putting 

dv  =  sin  X  dx ;     then  u  =  sin"~^  x, 

V  =  —  cos  X,  du  =  (?i  —  1)  sin''~^a.'  cos  x  dx. 

.:    I  sin''xdx  =  —  sin"~^ic  cos  x  -\-  (n  —  1)  |  sin''~-cc  cos^;c  dx 

=  —  sin'^-^a?  cos  x  -\-  (n  —  1)  j  sin''~-ic  (1  —  sin- a.')  dx 

=  —  sin"~^cc  cos  X  +  (n  —  1)  j  sin"'~^xdx  —  (n  —  1)  I  sin^'xdx. 

From  this,  on  transposition  and  division  by  n, 

/'   n    ^            si n"^^  X  COS  a;  ,  n  —  1  f  ■   ,,o     7  /in 

sm"  xdx  = 1 —  I  sm"  -  x  dx.  (1) 

This  is  a  useful  formula  of  reduction  when  n  is  a  positive 
integer.  From  it  can  be  deduced  a  formula  which  is  useful  when 
the  index  is  a  negative  integer.     For,  on  transposition  and  division 

by      ~  "  ;  formula  (1)  becomes 

/.   ,,_o     -,        sin"~^  X  cos  X  ,       n      C  -   n    ^ 
sin"  -xdx  = 1 I  sm"  x  dx. 
71—1               71  —  IJ 

This  result  is  true  for  all  values  of  n,  and,  accordingly,  for 
n  =  JV+  2.     On  putting  N+2  for  71,  this  becomes 

/.   ^     ,        sin^+^  X  cos  X  ,  N-\-2  C  .   ,^,0     7  /o\ 

sin^ic  dx  = -^— -  I  sm^+-xdx.  (2) 

If  ^is  a  negative  integer,  say  —  m,  (2)  may  be  written 

/dx    _  _      1        cos  X        m  —  2  r    dx  ^r^. 

sin'^a;         m  —  1  sin"*~'a;      m  —  lJ  sin"'"^^ 


122.]  TRIGONOMETRIC  FUNCTIONS.  219 

In  the  above  way  calculate  the  following  integrals  : 

Ex.1.    (1)    isin^xdx,     (2)    iam^xdx,     (3)    isin^x^x,     (4)    isin^xdx. 

Ex.  2.    (1)    f-:^,      (2)    f  .^^-,      (3)    f^^. 
^  ^  Jsin-^a;      ^      Jsin^x      ^      Jsin^x 

Ex.  3.  Compare  the  results  in  Exs.  1,  2,  with  those  obtained  for  these 
hitegrals  by  methods  of  the  preceding  articles. 

B.  Similarly  to  A  there  can  be  deduced  results  69,  71,  page  407, 
for  B.  Formula  69  is  useful  for  positive  indices,  and  71  for 
negative  indices. 

Ex.  4.   Deduce  formulas  69  and  71. 

Ex.  5.    (1)    (cos^xdx,     (2)    fcos^x^x,     (3)    (*-^^,      (4)    f-^. 
J  J  Jcos*x  J  cos^x 

Compare  results  with  those  obtained  by  methods  of  preceding  articles. 
(b)   To  find   I  sec'^xdx  ivJien  n  is  a  positive  integer  greater  than  1. 

Put  sec"^  xdx  =  dv;  then  sec"~^a:  =  w, 

tan  X  =  V,     (n  —  2)  sec"~^  x  tan  x  dx  =  du. 
.'.    I  sec'*  X  dx  =  sec"~^ x  tan  x  —  (n  —  2)  I  sec""^ x  tan^x  dx. 

From  this,  on  substituting  sec'a;  — 1  for  tan-ic,  and  solving 
for    I  sec"  a;  da;, 

/„     7        sec^^^x'tano;  ,  n  —  2  C     „_<,     , 
sec"  X  dx  =  — H I  sec"  -  x  dx. 
n  ~1            n  —  lJ 

Similarly,  result  73  for   |  cosec"  x  dx  can  be  obtained. 

Ex.  6.    (1)    fsec^xdx,     (2)    Tsec^xdx,     (3)  sec^xc^x. 

Ex.  7.    (1)    fcsc^xf^x,     (2)    fcsc*x(?x,     (3)  csc^x^Zx. 
Ex.  8.    Derive  formula  73. 

Ex.9.   From  formulas  72  and  73  derive  formulas  for    jsec«X(?x  and 
1  cosec*'  X  dx  which  are  applicable  when  n  is  a  negative  integer. 
[Suggestion  :  Use  method  employed  in  deducing  formulas  70  and  71.] 


220  INFINITESIMAL   CALCULUS.  [Ch.  XIII. 

(c)  To  find  I  taiV^xdx,  in  ivhich  n  is  a  positive  integer  greater 
than  1. 

I  tan**  xdx  =  I  taii"~^  x  tan^  xdx—  |  tan"~^  x  (sec-  x  —  1)  dx 

=  Aaii'^-^  X  d  (tan  x)  —  ftan'-^  x  dx 

^tanr^_  r^^^n-.^ax. 
n-1       J 

Similarly  can  be  shown  result  75  for   |  cot'' xdx. 

When  n  is  negative,  say   —  m,  then    j  tan"  xdx=   I  cot"*  x  dx, 

and    I  tan"* a;  da;  can  be  expressed  in  cotangents  by  formula  75. 

Formulas  applicable  to  cases  in  which  n  is  negative,  can  be 
deduced  from  formulas  74  and  75,  by  the  method  used  in 
deducing  formulas  70  and  71. 

Ex.  10.  Deduce  Formula  75,  and  formulas  applicable  to  ltaii«xd!x  and 
(  cot^xdx  when  n  Is  negative. 

Ex.  11.    (1)    (tsin^xdx,     (2)  cot*  x  dfx,     (3)    ftan*xc?x,     (4)    (tdni^xdx. 

{d)     j  sm*"a?cos**i»e?a?.     When  m  and  n  are  integers,  reduction 

formulas  can  be  derived  for  this  integral  in  a  manner  similar  to 
that  used  in  Art.  118 ;  that  is,  by 

(i)    Connecting  it  with  each  of  the  four  integrals  in  turn,  viz. : 
j  sin"'~^xcos'*a;da;,  j  sin"*a;cos"~^a;rZa;, 

j  sin™+^  a;  cos"  a?  cZo;,  j  sin"*a7C0S"+^a;dx; 

(ii)  Forming  a  new  function  by  giving  sin  x  and  cos  x  each  an 
index  one  greater  than  the  lesser  of  its  indices  in  the  required 
integral  and  the  integral  with  which  it  is  connected,  and  taking 
the  product ; 

(iii)  Giving  the  connected  integral  and  this  newly  formed 
function  each  an  arbitrary  coefficient,  and  equating  their  sum  to 
the  required  integral  j 


122.]  TRIGONOMETRIC  FUNCTIONS.  221 

(iv)  Determining  the  value  of  these  coefficients  by  proceeding 
as  in  Art.  118. 

The  derivation  of  these  reduction  formulas  is  left  as  an  exercise 
for  the  student ;  they  are  given  in  the  set  of  integrals,  Nos.  76-79.* 

Ex.  12.    Deduce  formulas  Nos.  7G-79  by  the  methods  outlined  above. 
Ex.  13.    Deduce  the  formulas  in  Ex.  12  by  integrating  by  parts. 
Ex.  14.    Apply  these  formulas  to  finding  the  following  mtegrals  : 

(1)    rsin2a-.cos2a:(Zx;    (2)    f  cos*  x  sin2  a; ;    (3)    C^^^dx. 
J  J  J  sin2  X 

•    Ex.  15.    Deduce  the  integrals  in  Ex.  14  by  the  method  outlined  in  {d). 

Note  1.     When  w  +  w  is  a  negative  even  integer,  the  above  integral  can 

be  expressed  in  the  form    i  /(tan  a:)cZ(tan  x). 

Ex.  16.     r^Ei?  dx  =  r?llli^  .  -J—  .  dx  =  ftan^  x  sec*  x  dx 

J  COS^  X  J  COS^  X      COS*  X  J 

=  \  tan^  X  (1  4-  tan2  x)  c?  tan  x  =  ^^(6  +  4  tan2  x)  tan*  x. 

Ex.17.    (1)   f^^^cZx,     (2)    (^^^dx,     (3)   f^-^^dx. 
J  sin^  X  J  cos^  X  J  sin*5  x 

Note  2.  Special  forms.  Integrals  80-87  are  occasionally  required.  For 
their  deduction  see  Murray,  Integral  Calculus,  Arts.  54-57,  or  other  texts. 
It  will  be  a  good  exercise  for  the  student  to  try  to  deduce  these  integrals  him- 
self. For  a  fuller  discussion  of  the  integration  of  irrational  and  trigonometric 
functions  see  the  article  Infinitesimal  Calculus  (Ency.  Brit.,  9th  edition), 
§§  124  on  ;  also  see  Echols,  Calculus,  Chap.  XVIII. 

Note  3.    On  integration  by  infinite  series.    See  Art.  126. 

Note  4.  Elliptic  integrals.  Elliptic  functions.  The  algebraic  inte- 
grands considered  in  this  book  give  rise  only  to  the  ordinary  algebraic, 
circular,  and  hyperbolic  t  functions.  (The  two  last  named  are  singly  periodic 
functions.)  Certain  irrational  integrands  give  rise  to  a  class  of  functions 
treated  in  hi<?her  mathematics,  viz.  the  elliptic  (or  doubly  periodic)  functions. 
The  term  elliptic  functions  is  somewhat  of  a  misnomer;  for  the  elliptic 
functions  are  not  connected  with  an  ellipse  in  the  same  way  as  the  circular 
functions  are  connected  with  the  circle,  and  the  hyperbolic  functions  with 
the  hyperbola.  The  elliptic  integrals  derived  their  name  from  the  fact  that 
an  integral  of  this  kind  appeared  in  the  determination  of  the  length  of  an 
arc  of  the  ellipse.     Out  of  the  study  of  the  elliptic  integrals  arose  the  modern 

*  These  formulas  are  derived  in  Murray,  Integral  Calculus,  Art.  51,  and 
Appendix,  Note  C.     Also  see  Edwards,  Integral  Calculus,  Art.  83. 
t  See  Appendix,  Note  A. 


222  INFINITESIMAL    CALCULUS.  [Ch.  XIII. 

extensive  and  important  subject  of  elliptic  functions  ;  this  accounts  fo/tlie 
term  elliptic  in  the  name  of  these  functions.  'J'he  student  may  take  a  glance 
forward  and  extend  his  mathematical  outlook  by  inspecting  Art.  174,  Note  4  ; 
Cajori,  History  of  Mathematics,  pages  279,  347-354 ;  the  section  on  elliptic 
integrals  in  the  article  mentioned  in  Note  2,  in  particular,  §§  191,  192,  204, 
205,  206,  219,  220  ;  W.  B.  Smith,  Infinitesimal  Analysis,  Vol.  I.,  Arts.  123-125 ; 
Glaisher,  Elliptic  Functions,  pages  6,  175,  etc. 

EXAMPLES. 

1.  Derive  integrals  Nos.  80-82,  85-87. 

2.  Derive  several  of  the  integrals  18-30,  36-46,  53-65. 

Vt    ■,,  /ox    C      vdv  /ox    C  dx 


(1)  (-^dt.      (2)  (—^^-^ —      (3)  r '- 


Vl  -  4  x^ 


(4)    r ^^-^  (5)   ^^""-"^"dx.  (6)    (   (2^+1)^^-^    . 

-^  (1  +  ic2)  Vl  -  4  x-^  ^^  ^  Vx-!  +  3  a;  +  5 

(~)    C     ('■^x-j-l)dx  ,gN    r dv .g.    r xdx 

J^Vx2  +  3x  +  5'  J(^'+l)Vt^M^"  J  (x^- 10)3- 

(10)    f — ^ (11)    f ^^'  (12)    f  ^^''^"'^'^da;. 

^  ^  J  (x2  +  4)3         ^  ^  J  (1  +  x-^)  vr^^  ^     ^' 


4.    Derive  the  following  integrals 


dx  =  a  sin-i  —  Va'^  —  ic^. 


(1)  fV^^i^ 

(2)    f  J^±^  dx  =  -  V(a  +  x)(&-x)  -  (a  +  &)sin-i  J- 
J  >!?)  —  X  >a 


+  6 


(3)  rJ^ — ^c?a;  =  V(a-x)(6  +  x)  +  (a  +  6)sin-iJ?-+|. 

(4)  (*  J^dii?  ax  =  V(a  +  x)(6  +  x)  +  (a  -  6) log (  VoT^  +  v 6Tx) . 
^    '6  +  X 

(5)  f  ^^  -..2sin-iJ^HJ. 
-^  V(x-a){b-x)  ^b-a 

5.  Show  that,  if /(i<,  v)  is  a  rational  function  of  7i  and  v,  and  m  and  w  are 

m 

integers,  then  /{x^,  (a  +  bx'^)>^}xdx  can  be  rationalised  by  means  of  the  sub- 
stitution a  +  &x2  =  ;s".  (Ex.  14,  or  Note  3,  Art.  117,  is  a  particular  case  of  this 
theorem.)  ^ 

6.  Show  that  (1)   r  sin2- dx  =  1  •  ^  •  5  ...  (2m- 1)  .  ^ 

Jo  2. 4. 6...  2  m         2 


(^)  r 


2.  4.  6...  2  m         2 

sin2"»+ixdx  =  —  —  (m  being  an  integer). 

3.5.7...  (2w  +  l) 


CHAPTER   XIV. 


APPROXIMATE   INTEGRATION. 
INTEGRATION. 


MECHANICAL 


123.  Approximate  integration  of  definite  integrals.     It  has  been 

shown  in  Arts.  95,  96,  98,  that :  (a)  the  definite  integral  I  f{x)dx 

may  be  evahiated  by  finding  the  anti-differential  of  f(x)dx,  <f>(x) 
say,  and  calculating  cfj(b)  —  <^(a) ;  (b)  this  last  number  is  also  the 
measure  of  the  area  of  the  figure  bounded  by  the  curve  y  =f(x), 
the  a>axis,  and  the  two  ordinates  for  which  x  =  a  and  x  =  b.  In 
only  a  few  cases,  however,  can  the  anti-differential  of  f{x)dx  be 
found ;  in  other  cases  an  approximate  value  of  the  definite  inte- 
gral can  be  obtained  by  making  use  of  fact  {b).  Thus,  on  the  one 
hand  the  evaluation  of  a  definite  integral  serves  to  give  the 
measurement  of  an  area;  on  the  other  hand  the  accurate  measure- 
ment of  a  certain  area  will  give  the  exact  value  of  a  definite 
integral,  and  an  approximate  determination  of  this  area  will  give 
an  approximate  value  of  the  integral.  The  area  described  above 
may  be  found  approximately  by  one  of  several  methods ;  two  of 
these  methods  are  explained  in  Arts.  124  and  125. 

124.  Trapezoidal  rule  for  measuring  areas  (and  evaluating  definite 

integrals).      Let  the  value  of  the  definite  integral    I  f(x)dx  be 

required.  Plot  the  curve 
y=f{x)  from  ic  =  a  to  x=b. 
Let  OA  =  a,  OB  =  b,  and  draw 
the  ordinates  AP  Siud  BQ.  By 
Art.  96,  the  measure  of  the 
area  APQB  is  the  value  of  the 
required  integral.  An  approxi- 
mate value  of  the  area  APQB 

can  be  found  in  the  following  Fig.  63. 

223 


B  X 


224  INFINITESIMAL    CALCULUS.  [Ch.  XIV. 

way.  Divide  the  base  AB  into  n  intervals  each  equal  to  Ax,  and 
at  the  points  of  division  A^,  ^2?  ^3>  ••*?  erect  ordinates  A^P^^ 
A2P2,  A^Ps,  •••.  Draw  the  chords  PI\,  P^P.,,  P2A,  •',  thus 
forming  the  trapezoids  AP^,  AiPo,  A.2P2,  •••.  The  sum  of  the 
areas  of  these  trapezoids  will  give  an  approximate  value  of 
the  area  of  APQB. 

ATe3iAP,  =  ^(AP-^AiP,)Ax, 

area  A1P2  =  ^  (^lA  +  A2P2)  ^x, 

area  AoP^  =  -J-  (AP2  +  A^Ps)  Ax, 


area  A,_,Q  =  1  (^„_iP,_i  +  BQ)  \x. 

.'.  area  of  trapezoids  =  (i  AP  +  A^P^^  +  A2P2  H -f-  A-i^n-i 

+  iBQ)Ax. 
This  result  may  be  indicated  thus  :    . 

area  trapezoids  =(|  +  l  +  l  +  ...  +  l+|) Aac, 

in  which  the  numbers  in  the  brackets  are  to  be  taken  with  the 
successive  ordinates  beginning  with  AP  and  ending  with  BQ. 

Note.  It  is  evident  that  the  greater  the  number  of  intervals  into  which 
6  —  a  is  divided,  the  more  nearly  will  the  total  area  of  the  trapezoids  come 
to  the  actual  area  between  the  curve  and  the  x-axis,  and,  accordingly,  the 
more  nearly  to  the  value  of  the  integral.    See  Exs.  1,  2. 

EXAMPLES. 

1.  Find  \    x^dx,  dividing  12  —  1  into  11  equal  intervals. 
Here  each  interval,  Ax,  is  1.     Hence,  approximate  value 

=  (^  .  12  +  22  +  32  +  42  +  52  +  6-2  +  72  +  82  -f  92  +  102  +  IP  +  1  .  122)  =  577|. 

/•I2  r^3  -|   12 

The  value  of  I  x^dx=  \—+  c\  =  575 1.  The  error  in  the  result  ob- 
tained by  the  trapezoidal  method  is  thus,  in  this  instance,  less  than  one- 
third  of  one  per  cent. 

2.  Show  that  if  22  equal  intervals  be  taken  in  the  above  integral,  the 

approximate  value  found  is  576.125. 

ri') 
8.    Show   that   on   using   the    trapezoidal    rule    for    evaluating  I    x'^dx^ 

if  10  Intervals  be  taken,  the  result  is  If  units  more  than  the  true  value, 
and  if  20  intervals  be  taken,  the  result  is»j\  of  a  unit  more  than  the  true 
value. 


125.] 


PARABOLIC  BULE. 


225 


4.  Explain   why  the   approximate  values    found    for    the    integrals   in 
Exs.  1,  2,  3,  are  greater  than  the  true  values. 

5.  Evaluate  I      cos  x  dx  by  the  trapezoidal  rule,  taking  10'  intervals. 

^^°         {Ans.    .0148.    The  calculus  method  gives  .0149.) 

/•320 

6.  Evaluate   \      sin  x  dx,  taking  30'  intervals. 

•^^^°  {Ans.    .0506.     Calculus  gives  .0508.) 

/•380 

7.  Evaluate  i      cos  x  dx,  taking  1°  intervals. 

•^^^°  (Ans.    .1509.     Calculus  giv6s.  1510.) 

125.  Parabolic  rule*  for  measuring  areas  and  evaluating  definite 
integrals.  Let  the  area  and  the  integral  be  as  specified  in  Art. 
124.  For  the  application  of  the  parabolic  rule,  the  interval  AB 
is  divided  into  an  even  y^ 
number  of  equal  intervals 
each  equal  to  Aa;,  say.  The 
ordinates  are  drawn  at  the 
points  of  division.  Through 
each  successive  set  of  three 
points  (P,  P„  P,),  (A,  Ps, 
P4),  •••,  are  drawn  arcs  of 
parabolas  whose  axes  are 
parallel  to  the  ordinates.  The  area  between  these  parabolic  arcs 
and  the  ic-axis  will  be  approximately  equal  to  the  area  between 
the  given  curve  and  the  x-axis.  The  area  bounded  by  one  6f  these 
parabolic  arcs  and  the  a>axis,  and  a  pair  of  ordinates,  say  the 
area  of  the  parabolic  strip  APP1P2A2,  will  now  be  found. 

Parabolic  strip  APP1P2A2  =  trapezoid  APP2A2  +  parabolic 

segment  PP1P2.  (1) 

Now  the  parabolic  segment  PP1P2 

=  two-thirds  of  its  circumscribing 
parallelogram  PPPaP.t        (2) 


Ax   A-i  A-i,  A^ 
Fig.  G4. 


*  This  rule,  which  is  much  used  by  engineers  for  measuring  areas,  is  also 
known  as  Simpson's  one-third  rule,  from  its  inventor,  Thomas  Simpson 
(1710-1761),  Professor  of  Mathematics  at  Woolwich. 

t  See  Art.  Ill,  Ex.  19. 


226  INFINITESIMAL    CALCULUS.  [Ch.  XIV. 

Area  trapezoid  APP^A^  =  \  AA2  (AP  -f-  A2P2)  ; 

area  PPP2P2  =  area  J.P'P'2^2  —  area  APP2A2 
=  2'^AA2'A,F^-iAA2(AP 

+  A2P2).  (3) 

Hence,  by  (1),  (2),  and  (3),  area  parabolic  strip  APP1P2A2 

=  (.4P+4AA  +  ^2/^2)y- 
Similarly,  area  of  next  parabolic  strip  ^^^2^^4-44 

=  (A2P2-\-4:A,P,  +  A,P,)^^-', 

and  so  on.  Addition  of  the  successive  areas  gives  total  area  of 
parabolic  strip  =(AP^^AP,  +  2  A2P2  +  4  A.P, 

+  2A^,+  -4-^Q)y- 
This  result  may  be  indicated  thus : 

Total  parabolic  area  =  (1+4+2  +  4  +  . ..  +  2  +  4  +  1)^^,  (4D 

o 

in  which  the  numbers  in  the  brackets  are  understood  to  be  taken 
with  the  successive  ordinates  beginning  with  AP  and  ending 
with  BQ. 

EXAMPLES. 

/•lO 

1.    Find  i     x^  (Za;,  taking  10  equal  intervals. 
Here,  each  interval  =  1.     Hence,  the  result  by  (4) 

=  (1  .  03  +  4  •  13  +  2  •  23  +  4  .  33  +  2  .  43  +  4  .  53  +  2  .  63  +  4  .  73 
+  2  .  83  +  4  .  93  +  1  .  103)  X  1  =  2500. 


t/y.4         nio 
—  +  c       =  2500. 


2.  Calculate  the  above  integral,  using  the  trapezoidal  rule  and  taking 

10  equal  intervals. 
rn 

3.  Evaluate   i    x'^  dx,  both  by  the  trapezoidal  and  the  parabolic  rules, 

taking  10  equal  intervals. 

4.  Evaluate  Ex.  1,  Art.  124,  by  the  parabolic  rule.     Why  is  the  result 
the  true  value  of  the  integral  ? 

5.  Show  that  there  is  only  an  error  of  14  in  20,000  made  in  evaluating 
\    x'^dx  by  the  parabolic  method,  when  10  intervals  are  taken. 


126.]  INTEGBATION  IN  SEBIES.  227 

6.  Find  the  error  in  the  evakiation  of  the  integral  in  Ex.  5  by  the 
trapezoidal  method,  when  10  intervals  are  taken. 

7.  Evaluate  the  integrals  in  Exs.  6,  7,  Art.  124,  by  the  parabolic  rule. 

Note.  For  a  comparison  between  the  trapezoidal  and  parabolic  rules, 
for  a  statement  of  Durand's  rule,  which  is  an  empirical  deduction  from 
these  two  rules,  for  a  statement  of  other  rules  for  approximate  integration, 

and  for  a  note  on  the  outside  limits  of  error  in  the  case  of  the  trapezoidal 
and  parabolic  rules,  see  Murray,  Integral  Calculus,  Arts.  86,  87,  Appendix, 
Note  P],  and  foot-note,  page  186. 

126.   Integration  in  series.     The  methods  described  or  referred 

to  in  Arts.  124  and  125  for  evaluating  a  definite  integral   |  f(x)d^, 

give  a  numerical  result  only,  and  do  not  convey  any  information 
as  to  the  anti-differential  of  f(x)dx.  Some  information,  however, 
about  the  anti-differential  of  f{x)dx  can  be  obtained  in  certain 
cases  by  expanding  f(x)  in  a  series  in  ascending  or  descending 
powers  in  x  and  then  integrating  f(x)dx  term  by  term.  The 
expanded  series  can  represent  f(x)  only  for  the  values  of  x  in  a 
certain  definite  range,  namely,  the  range  of  values  for  which  the 
series  is  convergent.  The  series  obtained  by  integration  is  con- 
vergent for  the  same  range  of  values  of  x,  and  /or  values  of  x 
in  this  range  represents  the  anti-differential.  See  Chapter  XIX. 
(in  particular,  Arts.  172,  174),  where  the  question  of  integration 
in  series  is  more  fully  discussed.  The  following  examples  and 
note  are  given  here  mainly  for  the  purpose  of  drawing  attention 
to,  and  arousing  interest  in,  questions  relating  to  series. 

EXAMPLES. 

1.  Given  that  e»  =  l  +  a:-f-  —  +  ?-+  •••,  show  that  \  e""  dx  —  e''  -\-  c, 
in  which  c  is  a  constant.  ■      ^  ' 

2.  Given    that    cos  x  =  \  —  —  -f^ ...,    and    that   sin  a:  =  x  —  —  -|-  — 

2!4!  3!5! 

—  •••,  show  that  \  cos  x  dx  =  sin  x  +  c,  and  that  j  sin  x  (?x  =  —  cos  x  +  c. 

3.  Do  Ex.  2,  Art.  174. 

4.  Find  an  approximate  value  of  the  area  of  the  four-cusped  hypocycloid 
inscribed  in  a  circle  of  radius  8  inches.  (This  area  can  also  be  found  exactly ; 
see  Art.  137,  Note  5,  Ex.) 


228  INFINITESIMAL   CALCULUS.  [Ch.  XIV. 

Note.  Expansion  of  functions  in  series:  (a)  by  differentiation; 
(6)  by  integration.  For  remarks  on  this  topic  and  for  the  warrant  for  the 
operations  in  Exs.  5,  6,  7,  see  Chapter  XIX.,  in  particular,  Arts.  168(e), 
172,  173,  174. 

5.  Suppose  it  is  known  that  sin  y,  =  X —  —  + ^ •••,  (1) 

3  !      6 ! 

and  thut  —  (sin  x)  =  cos  x. 

Differentiation  of  the  members  in  (1)  gives 

cosx  =  l-^  +  ^-.... 
2  !     4  ! 

6.  By  the  binomial  theorem, 

(1  -  x2)^  =  1  _  I  x2  -  1  a;4  -  ^i^ccfi .  (1) 

Differentiation  of  each  member  of  (1)  and  division  by  %  gives 

(1  -  a:2)~^  =  i-f  ^  x2  +  f  a:*  +  t\  »^^  +  ••••  (2) 

Result  (2)  may  be  verified  by  expanding  (1  —  x^)"^^  by  the  binomial 
theorem. 

7.  Given     that    d(tan-ix)= — ^     (Art.    51)=l-x2  +  x* ,    find 

tan-ix.  -^^^^^ 

i  r^      x^ 

On  mtegration,  tan-i  x-\-  c  —  x V- —  ••-. 

3      5 

From  this,  on  putting  x  =  0,  tan-i  0  +  c  =  0.     .-.  c  =  —  tan-i  0  =  ±  nv^  in 

which  n  is  any  integer. 

.-.  tan-i  X  =  WTT  +  X  -  —  +  — . 

8.  Do  Exs.  3-5,  8,  9,  Art.  174. 

127.  Mechanical  devices  for  integration.  The  value  of  a  definite 
integral  may  be  determined  by  various  instruments.  Accordingly, 
they  may  be  called  mechanical  integrators.  Of  these  there  are 
three  classes,  viz.  planimeters,  integrators,  and  integraphs.  These 
instruments  are  a  great  aid  to  civil,  mechanical,  and  marine 
engineers.  The  area  of  any  plane  figure  can  be  easily  and  accu- 
rately calculated  by  each  of  these  mechanisms.  Their  right  to  be 
termed  mechanical  integrators  depends  on  the  facts  emphasised  in 
Arts.  96,  98,  123-125;  the  facts,  namely,  that  a  definite  integral 
can  be  represented  by  a  plane  area  such  that  the  number  of  square 
units  in  the  area  is  the  same  as  the  number  of  units  in  the  inte- 
gral, and  hence  that  one  way  of  calculating  a  definite  integral  is 
to  make  a  proper  areal  representation  of  the  integral  and  then 
measure  this  area. 


127.]  PLANIMETERS,  INTEGRAPHS.  229 

Planimeters,  which  are  of  two  kinds,  viz.  polar  planimeters  and 
rolling  planimeters,  are  designed  for  finding  the  area  of  any  plane 
surface  represented  by  a  figure  drawn  to  any  scale.  The  first 
planimeter  was  devised  in  1814  by  J.  M.  Hermann,  a  Bavarian 
engineer.  A  jjolar  planimeter,  which  is  a  development  of  the 
planimeter  invented  by  Jacob  Amsler  at  Konigsberg  in  1854,  is 
the  one  most  extensively  used.  By  it  the  area  of  any  figure  is 
obtained  by  going  around  the  boundary  line  of  the  figure  with 
a  tracing  point  and  noting  the  numbers  that  are  indicated  on  a 
measuring  wheel  when  the  operation  of  tracing  begins  and  ends. 

Integrators  and  integraphs  also  serve  for  the  measurement  of 
areas  ;  they  are  adapted,  moreover,  for  making  far  greater  compu- 
tations and  solving  more  complicated  problems,  such  as  the  calcu- 
lation of  moments  of  inertia,  centres  of  gravity,  etc.  The  integraph 
(see  xirt.  100,  Notes  2,  3)  is  the  superior  instrument,  for  it  directly 
and  automatically  draws  the  successive  integral  curves.  These 
give  a  graphic  representation  of  the  integration,  and  are  of  great 
service,  especially  to  naval  architects.  The  measure  of  an  ordi- 
nate of  the  first  integral  curve,  when  multiplied  by  a  constant 
belonging  to  the  instrument,  gives  a  certain  area  associated  with 
that  ordinate  (see  Art.  100). 

Note  1.  A  bicycle  with  a  cyclometer  attached  may  be  regarded  as  a 
mechanical  integrator  of  a  certain  kind ;  for  by  means  of  a  self-recording 
apparatus  it  gives  the  length  of  the  path  passed  over  by  the  bicycle. 

Note  2.  Planimeters  and  integrators  are  simple,  and  it  is  easy  to  learn 
to  use  them. 

Note  3.  A  brief  account  of  the  planimeter^  references  to  the  literature  on 
the  subject,  and  a  note  on  the  fundamental  theory,  will  be  found  in  Murray, 
Integral  Calculus^  Art.  88,  and  Appendix,  Note  F.  Also  see  Lamb,  Calculus^ 
Art.  102  ;  Gibson,  Calculus,  §  130.  For  a  fuller  account  see  Henrici,  Report 
on  Planimeters  (Report  of  Brit.  Assoc,  for  Advancement  of  Scienfee,  1894, 
pages  496-523) ;  Hele  Shaw,  Mechanical  Integrators  (Proc.  Institution  of 
Civil  Engineers,  Vol.  82,  1885,  pages  75-143).  For  references  concerniDg 
the  integraph  see  Art.  100,  Note  3. 

N.B.  Interesting  information  concerning  planimeters,  integrators,  and 
the  integraph,  with  good  cuts  and  descriptions,  are  given  in  the  catalogues  of 
dealers  in  drawing  materials  and  surveying  instruments. 


CHAPTER   XV. 

SUCCESSIVE    INTEGRATION.      MULTIPLE    INTEGRALS. 
APPLICATIONS. 

128.  In  Chapter  VI.  (see  Arts.  68,  69,  70),  successive  deriva- 
tives and  differentials  of  functions  of  a  single  variable  were 
obtained.  In  Chapter  VIII.  (see  Arts.  79,  80,  82),  successive  par- 
tial derivatives  and  partial  differentials  of  functions  of  several 
variables  were  discussed.  In  this  chapter  processes  which  are  the 
reverse  of  the  above  are  performed  and  are  employed  in  practical 
applications. 

129.  Successive  integration :  One  variable.     Applications. 

Suppose  that  1 /(i«)c?^  =/i(^)?  (1) 

fA(x)dx=f,(x),  '  (2) 

jMx)dx=Mx).  (3) 

Then,  by  (3)  and  (2),        fs(x)=j\^ff,{x)dx~\dx',  (4) 

By  (4)  and  (1),  /.(x)  =f\f(^ff(^)^^)  ^^]  ^^-      (^) 

This  k  written  f^(x)  =  f  f  ff{x)  {dxf, 

or,  more  usually,  /3(^)  =  1    I    \  f{x)dx^.  (6) 

The  second  member  of  (6)  is  called  a  triple  integral.    Similarly, 

the  second  member  in  (4)  is  usually  written  I    |  f^(x)dx^,  and  is 
called  a  double  integral. 

In  general,  I   I   I  "•  j  f(3o)dac^  denotes  the  lesult  obtained  by 


230 


128,  129.]  SUCCESSIVE  INTEGRATION.  231 

integrating  f(x)dx  n  times  in  succession.  This  integral  is  indefi- 
nite unless  end  values  of  the  variable  be  assigned  for  each  of  the 
successive  integrations.  This  integral  and  the  integrals  in  (4)  and 
(5)  are  called  multiple  integrals. 

Note.    It  should  be  observed  that  here  dx"  denotes  dxdxdx-"io  n  factors, 
i.e.  {dxy,  and  not  d  •  x»  {i.e.  nx'^-^dx).     [Compare  Art.  70.] 

EXAMPLES. 
1.   Find  r  r  (x'^dx^. 


=  ^  4-  kix"  +  cax  +  f3  ; 


for,  since  Ci  is  an  arbitrary  constant,  ^  may  be  denoted  by  an  arbitrary  con- 


2 
stant  k\. 


195 


■  -  ri>''^-r[j>-]-=j;[f+^]>=fr-='-i 

3.  Determine  the  curves  for  every  point  of  which  -^-^  =  0.     Which  of 

these  curves  goes  through  the  points  (1,  2),  (0,  3)  ?     Which  of  these  curves 
has  the  slope  2  at  the  point  (3,  5)  ? 

TT  ^^V 

On  integrating,  -^  =  Ci. 

On  integrating  again,  y  =  cix  +  C2, 

which  represents  all  straight  lines. 

For  the  line  going  through  (1,  2)  and  (0,  3),  2  =  Ci  +  C2  and  3  =  0  +  C2  ; 
whence  ci  =  —  1,  C2  =  3.     Hence  the  line  isx  +  y  =  S. 

For  the  line  having  the  slope  2  at  (3,  5),  Ci  =  2  and  5  =  3  Ci  +  C2,  whence 
C2  =  —  1.     Hence  the  line  is  y  =  2x—  1. 

4.  Determine  the  curves  for  every  point  of  which  the  second  derivative 
of  the  ordinate  with  respect  to  the  abscissa  is  6.  Which  of  these  curves 
goes  through  the  points  (1,  2),  (-  3,  4)  ?  Which  of  them  has  the  slope  3  at 
the  point  (  -  2,  4)  ? 


232  INFINITESIMAL   CALCULUS.  [Ch.  XV. 

N.B.  The  student  is  recommended  to  write  sets  of  data  like  those  in 
Exs.  3-7,  and  determine  the  particular  curves  that  satisfy  them.  He  is  also 
recommended  to  draw  the  curves  appearing  in  these  examples. 

5.  Determine  the  curves  for  every  point  of  which  the  second  deriva- 
tive of  the  ordinate  with  respect  to  the  abscissa  is  6  times  the  number  of 
units  in  the  abscissa.  Which  of  these  curves  goes  through  the  points  (0,  0) 
(1,  2)  ?     Which  of  them  has  the  slope  2  at  (1,  4)  ? 

6.  Determine  the  curves  in  which  the  second  derivatives  -v4  from  point 

to  point  vary  as  the  abscissas.  Find  the  equation  of  that  one  of  these  curves 
which  passes  through  (0,  0),  (1,  2),  (2,  5).  Find  the  equation  of  that  one  of 
these  curves  which  passes  through  (1,  1),  and  has  the  slope  2  at  the  point 

(2,  4). 

7.  Determine  the  curves  in  which  the  second  derivative  of  the  abscissa 
with  respect  to  the  ordinate  varies  as  the  ordinate.  Which  of  these  curves 
passes  through  (0,  1),  (2,  0),  (3,  5)  ?  Which  of  them  has  the  slope  ^  at 
(1,  2),  and  passes  through  (—  1,  3)  ? 

8.  A  body  is  projected  vertically  upward  with  an  initial  velocity  of  1000 
feet  per  second.  Neglecting  the  resistance  of  the  air  and  taking  the  accelera- 
tion due  to  gravitation  as  32.2  feet  per  second,  calculate  the  height  to  which 
the  body  will  rise,  and  the  time  until  it  again  reaches  the  ground. 

9.  Do  Ex.  20,  Art.  68.  ^ 

10.  When  the  brakes  are  put  on  a  train,  its  velocity  suffers  a  constant 
retardation.  It  is  found  that  when  a  certain  train  is  running  30  miles  an 
hour  the  brakes  will  bring  it  to  a  dead  stop  in  2  minutes.  If  the  train  is  to 
stop  at  a  station,  at  what  distance  from  the  station  should  the  engineer 
whistle  "down  brakes"  ?     (Byerly,  Problems  in  Differential  Calculus.) 

130.   Successive  integration  :  several  variables.     Suppose  that 

J/ {^,  y,  ^) ^2;  =  Ux,  y,  2),  (1) 

ffi(^,  y,  ^)dy  =,f2(^,  y,  ^),  '  (2) 

jf2(x,  y,  z)  dx  =fi(x,  y,  z).  (3) 

The  integration  indicated  in  (1)  is  performed  as  if  y  and  x  were 
constant;  the  integration  in  (2)  as  if  x  and  z  were  constant;  the 
integration  in  (3)  as  if  z  and  y  were  constant.     (Compare  Arts,  79, 


130.]  SUCCESSIVE  INTEGRATION.  233 

From  (3)  and  (2),  fs(x,  y,  z)  =J\  j^f^(x,  y,  z)dy  \ dx;  (4) 

from  (4)  and  (1),  =J  |  flf-^^'"'  ^'  ^^  ^^1"^^  \  '^'^'  ^^^ 

The  second  member  in  (4)  is  often  written 

Jjfi(^,y,z)(il/d^;  (6) 

the  second  member  in  (5)  is  often  written 

ffffi^,  yy  2)  dz  dy  dx.  (7) 

The  integral  in  (6)  is  called  a  double  integral,  and  the  integral 
in  (7)  a  triple  integral. 

Note  1.  It  should  be  observed  that  according  to  (2),  (3),  and  (4),  inte- 
gral (6)  is  obtained  by  first  integrating /i(x,  y,  z)  with  respect  to  ?/,  and  then 
integrating  the  result  with  respect  to  x  ;  in  (7),  according  to  (1),  (2),  (3), 
and  (5),  the  first  integration  is  to  be  made  with  respect  to  z,  the  second  with 
respect  to  y,  and  the  third  with  respect  to  x.  That  is,  XhQ  first  integration  sign 
on  the  right  is  taken  icith  the  first  differential  on  the  left,  the  second  integra- 
tion sign  from  the  right  with  the  second  differential  from  the  left,  and  so  on. 
When  end-values  are  assigned  to  the  variables,  careful  attention  must  be  paid 
to  the  order  in  which  the  successive  integrations  are  performed. 

Note  2.  The  notation  used  above  for  indicating  the  order  of  the  variables 
with  respect  to  which  the  successive  integrations  are  to  be  performed,  is  not 
universally  adopted.  Oftentimes,  as  may  be  seen  by  examining  various  texts 
on  calculus  and  works  which  contain  applications  of  the  calculus,  integrals  (6) 
and  (7)  are  written 

\  ) /i(^^  y^  ^)  ^^  ^y^    Ml  •^^^'  ^'  ^^  ^^^  ^^y  ^^  respectively. 

In  this  notation  the  first  integration  sign  on  the  right  belongs  to  the  first 
differential  on  the  right,  the  second  integration  sign  from  the  right  to  the 
second  differential  from  the  right,  and  so  on  ;  and  the  integrations  are  to  be 
made,  first  with  respect  to  z,  then  with  respect  to  ?/,  and  then  with  respect  to  x. 
In  particular  instances,  the  context  will  show  what  notation  is  employed. 

EXAMPLES. 

1.  f  f  f ^ V^  ^^  (iy  ^^^  =  f  f^^^  (f  +  ^i)  ^y  ^^ 


234  INFINITESIMAL   CALCULUS.  [Ch.  XV. 

2.  CCC  v^yz^  dz  dy  dx  (i.e.  P"^*  f^T' f -T^^^^^  ^^  ^^^  ^^^ 

4  J2       L2         Ji  2       4  J2 

3.  ^;-j:'^Y-<l!>do.^=l'^-{£^'^Vdrj}.l.  =f  x3[|+  c];^<& 


iJVo 


x6-iK3)dx  =  28,Vo- 


4.   Evaluate  the  following  integrals  :  (1)    i    i'  \     xy'^z  dz  dy  dx. 
(2)    i      Y  {Zw-'lv^dw  dv.  (3)    j^  J      ^st  -  t^  ds  dt. 


'TT  ^a{l—coa0) 


r^  cos  ddr  dd. 


jr 
/»2    /•2/r  /•2a  cos  0 

(6)    J     j      \  r^  sin  d  dr  d(p  de. 

n 

J' 2    /•2a  cos  0 
0  X  '■*"''■ 


■'(£) 


rd^  (^r. 


V^ 


rdr  dd. 


♦131.   Application  of  successive  integration  to  finding  areas:   rec- 
tangular coordinates. 

EXAMPLES. 

1.   Find  the  area  between  the  curve  y"^  =  Sx,  the  x-axis,  and  the  ordinate 
for  which  x  =  S. 

At  P,  any  point  within  the  figure   OWM 
whose  area  is  required,  suppose  that  a  rectan- 
gle PQ  having  infinitesimal  sides  dx  and  dy 
parallel  to  the  axis  is  constructed.     The  area 
OTFJf  is  the  limit  of  the  sum  of  all  rectangles 
such  as  PQ  which  can  be  constructed  side  by 
side  in  OWM.     Let  one  of  the  vertical  sides  of 
the  rectangle  be  produced  both  ways  until  it 
meets  the  curve  and  the  aj-axis  in  T  and  S; 
complete  the  rectangle  TV  as  in  the  figure. 
First,  find  the  area  of  the  rectangular  strip  TV  by  finding  tlie  limit  of  the 
sum  of  the  rectangles  PQ  inscribed  in  it  from  >S'  to  T;  then  find  the  limit 
of  the  sum  of  the  strips  like  TV  which  can  be  inserted  between  OF  and  31 W. 


Fig.  65. 


131,132.]  SUCCESSIVE  INTEGBATION.  235 


Area  TV  =  lim  ^  (rectangles  PQ)  =  f         dy  dx  =  y/^x  dx.  (1) 

y&tS 

Area  0.¥Jr  =  lira  2^  (strips  TF)  =  J^  J  J  ^      <^1/ J  ^^  (2) 

X  atO 

=  2\/2  I    x^ dx  =  4VQ  square  units. 

The  last  expression  in  (2)  is  usually  written  (    i    ^dydx. 

The  area  of  O  WM  may  also  be  found  by  finding  the  limit  of  the  sum  of  the 
rectangles  PQ  which  may  be  inserted  between  R  and  U,  and  then  finding 
the  limit  of  the  sum  of  the  strips  like  PL  which  may  be  inserted  between 
03/ and  W.     Thus, 

area  PL  =  ^^^'^  dx  dy  =  £^  dx  dy  =  (^S-^^dy;  (3) 

area  OMP  =  J;;;(3  - 1')  <J,  =  {^'[3  -  |')  dy  =  iV6.  (4) 


Prom  (3)  and  (4),  areaOi¥P=l        I     ^^<^y 

Jo       Jy'^ 


8 

Note  1.     The  last  expression  in  (1)  is  y  dx,  the  element  of  area  employed 
in  Art.  96. 

Note  2.     Ex.  1  has  been  solved  as  above  merely  in  order  to  give  a  prac- 
tical application  of  double  integration. 

Note  3.     For  finding  areas  by  double  integration  in  the  case  of  polar 
coordinates,  see  Art.  136,  Note  3. 

2.  Express  some  of  the  areas  in  Art.  Ill  by  double  integrals,  and  per- 
form the  integrations. 

3.  Find  by  double  integration  the  area  included  between  the  parabolas 
3  y2  —  25  X  and  ox^  =  9y.     [See  Murray,  Integral  Calculus,  Art.  61,  Ex.  1.] 

132.   Application  of   successive   integration  to  finding  volumes : 
rectangular  coordinates. 

EXAMPLES. 

1.   Find  the  volume  bounded  by  the  surface  whose  equation  is 

a2"^62"^c2~   • 

Fig.  0-APC  represents  one-eighth  of  the  volume  required.     Suppose  that 
an  infinitesimal  parallelepiped  Pi  §3  is  taken  at  Pi  (a;,  y,  o).  having  infinitesi- 


236 


INFINITESIMAL   CALCUL US. 


[Ch.  XV. 


mal  sides  dXj  dy,  dz,  parallel  to  the  x-,  y-,  and  2;-axes,  respectively.  The 
volume  of  0-ABC  is  the  limit  of  the  sum  of  all  infinitesimal  parallelepipeds 
such  as  Pi §3  which  can  be  enclosed  by  OBA^  OAC,  OCB,  and  the  curvi- 


FiG.  66. 

linear  surface  ABC.  Construct  a  parallelepiped  PQi  by  producing  the 
vertical  faces  of  Pi  $3  to  the  height  Pi  P.  (The  point  P(ic,  y,  z)  is  taken  on 
the  surface  ABC.) 


/•«  at  P  I     /•«=«  vl— ^ —  I 

Vol.  P^i  =  J         dzdydx  =  \  J  «»    '*  dz  \dydx. 


(1) 


Note  1.  The  numbers  x  and  y  are  constant  along  PiP,  and,  accordingly, 
in  the  integration  of  (1)  x  and  y  are  treated  as  constants. 

Now  take  a  slice  BGL  the  planes  of  whose  faces  coincide  with  two  faces 
of  P^i,  as  shown  in  the  figure. 

Vol.  slice  BPGLS  =  limit  of  sum  of  parallelepipeds  P^i  from  S  to  G. 

That  is,     vol.  slice  BG  =  \  |  <'''    ^^  dz  \dy  -  dx 

Jy&tS    \_Jz=0  J 

1-x  /       ie*  r~  -V  /      **      y^       1  1 

Note  2.  The  number  x  is  constant  along  SG,  and,  accordingly,  in  the 
integration  of  (2)  x  is  treated  as  a  constant. 

Now  find  the  limit  of  the  sum  of  all  infinitesimal  slices  like  BGL  from 
OCB  to  A  ]  i.e.  from  x  =  0  to  x  =  a.     This  limit  is  the  volume  of  0-ABC. 


132.]  SUCCESSIVE  INTEGRATION.  237 

,.  vol.  0-ABC=("''\r'^'^'U' 


Vi---^ 


dy  \  dx 


=  jojo  jo  "^    *^^^^y^^.  (3) 

On  performing  the  integrations  indicated  in  (3)  (see  Ex.  4  (5),  Art.  130),  it 
will  be  found  that 

vol.  0-ABC  =  I  wabc.     Hence  vol.  ellipsoid  =  |  xabc. 

-^  .     .^  -  .  rxiktA     ryatG    fzatP 

Note  3.     Result  (3)  may  be  written   \  \  i         dxdydz. 

Jx&tO     Jy&tS    JzSitPi 

Note  4.  The  initial  element  of  volume  PiQi,  i.e.  dxdydz,  is  an  infinitesi- 
mal of  the  third  order  ;  the  parallelopiped  PQi  is  an  infinitesimal  of  the 
second  order  ;  the  slice  BGL  is  an  infinitesimal  of  the  first  order. 

Note  5.  Equally  well,  slices  may  be  taken  which  are  parallel  to  the 
iC0-plane  or  to  the  y^-plane. 

Note  6.  Instead  of  the  parallelopiped  PQi,  equally  well,  a  similar  paral- 
lelopiped can  be  taken  whose  finite  edges  are  parallel  to  the  y-axis,  or  to  the 
ic-axis. 

2.  Perform  the  integrations  indicated  in  Ex.  1. 

3.  Do  Ex.  1  by  taking  the  elements  in  the  ways  indicated  in  Notes  5 
and  6. 

4.  From  the  result  in  Ex.  1  deduce  the  volume  of  a  sphere  of  radius 
a.  Also  deduce  the  volume  of  this  sphere  by  the  method  used  in  Ex.  1. 
(Compare  with  the  methods  used  in  Art.  112,  Ex.  19  and  Note  3.) 

5.  Two  cuts  are  made  across  a  circular  cylindrical  log  which  is  20  inches 
in  diameter ;  one  cut  is  at  right  angles  to  the  axis  of  the  cylinder,  the 
other  cut  makes  an  angle  of  60°  with  the  first  cut,  and  both  cuts  intersect 
the  axis  of  the  cylinder  at  the  same  point.  Find  the  volume  of  each  of  the 
wedges  thus  obtained. 

6.  As  in  Ex.  5,  for  the  general  case  in  which  the  radius  of  the  log  is  a 
and  the  angle  between  the  cuts  is  a.     Thence  deduce  the  result  in  Ex.  5. 

7.  The  centre  of  a  sphere  of  radius  a  is  on  the  surface  of  a  right  cyl- 
inder the  radius  of  whose  base  is  -.  Find  the  volume  of  the  part  of  the 
cylinder  intercepted  by  the  sphere. 

8.  Taking  the  same  conditions  as  in  Exs.  5,  6,  excepting  that  the  cuts 
intersect  on  the  surface  of  the  log,  find  the  volume  intercepted  between  the 
cuts. 


238 


INFINITESIMAL   CALCULUS. 


[Ch.  XV. 


133.  Application  of  successive  integration  to  finding  volumes ; 
polar  coordinates. 

A.  The  use  of  polar  coordinates  in  finding  volumes  sometimes 
leads  to  easier  integrations  than  does   the   use   of   rectangular 

coordinates. 

Let  0,  the  origin,  be  taken 
as  pole.  The  infinitesimal  ele- 
ment of  volume  is  formed 
as  follows :  Take  any  point 
P{r,e,<^).  [Herer=OP,  6>  = 
angle  POZ,  <^  =  angle  XOM, 
0^  being  the  projection  of  OP 
on  XOY.  In  other  words, 
^  =  the  angle  between  the 
plane  XOZ  and  the  vertical 
plane  in  which  OP  lies.]  Pro- 
duce OP  an  infinitesimal  dis- 
tance dr  to  Pi,  and  revolve 
OPPi  through  an  infinitesimal 
le  dO  in  the  plane  ZOP  to  the  position  OQ.  Now  revolve 
OPPiQ  about  OZ  through  an  infinitesimal  angle  dcfi,  keeping 
$  constant.  The  solid  PP^QR  is  thus  generated.  Its  edges 
PPi,  PQ,  PR  are  respectively  dr,  rdd,  r  sin  0 deft;  its  volume  (to 
within  an  infinitesimal  of  an  order  lower  than  the  third)  is 
7^  sin  6  dr  d<^  dd.  On  determining  the  proper  limits  for  r,  <^,  0,  and 
integrating,  the  volume  required  is  obtained. 

Ex.  1.  Find  the  volume  of  a  sphere  of  radius  a,  using  polar  coordinates 
and  taking  0  on  the  surface  of  the  sphere  and  OZ  on  the  diameter  through  0. 
(It  will  be  found  that  the  volume  is  given  by  the  integral  in  Art.  130,  Ex.  4, 
(6).     See  Murray,  Integral  Calculus,  Art.  63,  Ex.  1.) 

B.  The  element  of  volume  can  be  chosen  in  another  way,  which 
sometimes  leads  to  simpler  integrations  than  are  otherwise  obtain- 
able.    An  instance  is  given  in  Ex.  2  below. 


Fig.  67 


EXAMPLES. 

2.   Another  way  of  doing  Ex.  7,  Art.  132. 

In  the  figure,  0-ABC  is  one-eighth  the  sphere,  and  the  solid  bounded  by 
the  plane  faces  ALBO,  AKO,  the  spherical  face  ALBVA,  and  the  cylindrical 


133.] 


SUCCESSIVE  INTEGRATION. 


239 


face  AVBOKA  is  one-fourth  of  the  part  of  the  cylinder  intercepted  by  the 
sphere. 

In  AOK  take  any  point  P. 
Let  0F  =  7\  and  angle  AOP=d. 
Produce  OP  an  infinitesimal  dis- 
tance dr  to  Pi,  and  revolve  OP  Pi 
through  an  infinitesimal  angle  dd. 
Then  PPi  generates  a  figure,  two 
of  whose  sides  are  dr  and  rdd. 
Its  area  (to  within  an  infinitesimal 
of  an  order  lower  than  the  second) 
is  r  dr  dd.  (See  Art.  136,  Note  3, 
Ex.  8.) 

On  this  infinitesimal  area  as 
a  base,  erect  a  vertical  column 
to  meet  the  sphere  in  M.  Then 
'PM  —  y/a^  —  r^,  and  the  volume 
of  the  column  is  Va^  —  r^  •  rdrdd. 
This  is  taken  as  the  element  of 
volume  ;  the  limit  of  the  sum  of  these  columns  standing  on  AOK  is  the  vol- 
ume required.  Keeping  6  constant,  first  find  the  limit  of  the  sum  of  the 
columns  standing  on  the  sector  extending  from  O  to  K  whose  angle  is  dd. 

/•  r=a  cos  0      . 

Since  0K=  acosd,  this  limit  is  \  Va^  —  r'^  •  rdrdd.     This  gives  the 

Jr=Ji 

volume  of  a  wedge-shaped  slice  whose  thin  edge  is  OB.  One-fourth  of  the 
volume  required  is  the  limit  of  the  sum  of  all  the  wedge-shaped  slices  of  this 
kind  that  can  be  inserted  between  AOB  and  COB ;  tliat  is,  from  ^  =  0  to 


Fig.  68. 


2' 


vol.  required  =  4 


Jd=0    Jr 


7-=a  cos  9 


\/a2 


r^  -rdrdd  =  ^ira^-^aK 

[See  Art.  130,  Ex.  4  (9).] 

In  this  instance  this  is  a  very  much  shorter  way  of  deriving  the  volume 
than  by  starting  with  the  element  dxdydz,  as  in  Art.  132. 

3.  Eind  the  volume  of  a  sphere  of  radius  a,  taking  O  at  the  centre : 
(1)  choosing  the  element  of  volume  as  in  ^  ;  (2)  choosing  it  as  in  B. 

4.  The  axis  of  a  right  circular  cylinder  of  radius  b  passes  through  the 
centre  of  a  sphere  of  radius  a  (a>  b).  Find  the  volume  of  that  portion  of 
the  sphere  which  is  external  to  the  cylinder. 


CHAPTER   XVI. 

FURTHER   GEOMETRICAL  APPLICATIONS   OF 
INTEGRATION. 

134.  In  this  chapter  the  calculus  is  used  for  finding  volumes 
in  a  particular  case,  for  finding  areas  of  curves  whose  equations 
are  given  in  polar  coordinates,  for  finding  the  lengths  of  curves 
whose  equations  are  given  either  in  rectangular  or  in  polar  coordi- 
nates, for  finding  the  areas  of  surfaces  in  two  special  cases,  and 
for  finding  mean  values  of  variable  quantities. 

N.B.  Many  of  the  problems  in  this  chapter  are  presented  in  a  general 
form.  In  such  cases  the  student  is  recommended,  when  he  obtains  the 
general  result,  to  make  immediate  application  of  it  to  particular  concrete 
cases. 

135.  Volumes  of  solids  the  areas  of  whose  cross-sections  can  be 
expressed  in  terms  of  one  variable.  In  Art.  112  the  volumes  of 
solids  of  revolution  were  found  by  making  cross-sections  of  the 
solid  at  right  angles  to  the  axis  of  revolution,  taking  these  cross- 
sections  an  infinitesimal  distance  apart,  and  finding  the  limit  of 
the  sum  of  the  infinitesimal  slices  into  which  the  solid  is  thus 
divided.  This  method  of  finding  the  volume  of  a  solid  can  some- 
times be  easily  applied  in  the  case  of  solids  which  are  not  solids 
of  revolution.  The  general  method  is  :  (a)  to  take  a  cross-section 
in  some  convenient  way;  (b)  to  express  the  area  of  this  cross- 
section  in  terms  of  some  variable ;  (c)  to  take  a  parallel  cross-sec- 
tion at  an  infinitesimal  distance  from  the  first  cross-section  ;  (d)  to 
express  the  volume  of  the  infinitesimal  slice  thus  formed,  in  terms 
of  the  variable  used  in  (b)  ;  (e)  to  find  the  limit  of  the  sum  of  the 
infinite  number  of  like  parallel  slices  into  which  the  solid  can 
thus  be  divided.  There  is  often  occasion  for  the  exercise  of  judg- 
ment in  taking  the  cross-sections  conveniently. 

240 


134,  135.] 


VOLUMES. 


241 


EXAMPLES. 

1.  Find  the  volume  of  a  right  conoid  with  a  circular  base  of  radius  a  and 
an  altitude  h. 

Note  1.  A  conoid  is  a  surface  which  may  be  generated  by  a  straight  line 
which  moves  in  such  a  manner  as  to  intersect  a  given  straight  line  and  a  given 
curve   and  always  be  parallel   to  a  . 

given  plane.  In  the  conoid  in  this 
example  the  given  plane  is  at  right 
angles  to  the  given  straight  line,  and 
the  perpendicular  erected  at  the 
centre  of  the  circle  to  the  plane  of 
the  base  intersects  the  given  straight 
line. 

Let  LM  be  the  fixed  line  and  ABB 
the  fixed  circle  having  its  centre  at 
C.  Take  a  cross-section  PQB  at 
right  angles  to  LM,  and,  accordingly, 
at  right  angles  to  a  diameter  AB. 
Let  it  intersect  AB  in  D,  and  denote 
CD  by  X. 

Area 

PQB  =:^PDQB  =  PD'  QD. 

Now  PD  —  h,  and,  by  elementary  geometry 


Fig.  69. 


QD  =  VAD'DB=  Via  -x)(a  +  x)  =  Va^  -  x^. 


.-.  area  PQB  =  hVa^  -  x^. 

Now  take  a  cross-section  parallel  to  PQB  at  an  infinitesimal  distance  from 
it.  Since  CD  has  been  denoted  by  x,  this  infinitesimal  distance  may  be 
denoted  by  dx. 

Vol.  LM-BQABB  =  2  vol.  LG-TSAT 

x&tA 

=  2  lim  (sum  of  slices  PQB) 


=  2  A  p  Va2  -  x-^  dx  =  l  TQ^h. 


That  is,  the  volume  of  the  conoid  is  one-half  the  volume  of  a  cylinder  of 
radius  a  and  height  h.     (See  Echols,  Calculus,  Ex,  3,  p.  266.) 

Note  2.  As  already  observed,  finding  the  volumes  of  solids  of  revolution 
is  a  special  case  under  this  article. 

Note  3.  Two  general  methods  of  finding  volumes  have  now  been  shown, 
namely,  the  method  shown  in  Arts.  132,  133,  and  the  method  shown  in  this 
article. 


242 


INFINITESIMAL   CALCULUS. 


[Ch.  XVI. 


2.  Do  Ex.  1,  denoting  AD  by  x. 

3.  Do  Ex.  8,  Art.  112  and  Ex.  1,  Art.  132  by  method  of  this  article. 

4.  Find  the  volume  of  a  right  conoid  of  height  8  which  has  an  elliptic 
base  having  semi-axes  6  and  4,  and  in  which  the  fixed  line  is  parallel  to  the 
major  axis.  Find  the  volume  in  the  general  case  in  which  the  height  is  /i, 
the  semi-major  axis  a,  and  the  semi-minor  axis  h. 

5.  A  rectangle  moves  from  a  fixed  point,  one  side  varying  as  the  dis- 
tance from  the  point,  and  the  other  side  as  the  square  of  this  distance.  At 
the  distance  of  3  feet  the  rectangle  is  a  square  whose  side  is  5  feet.  What 
is  the  volume  generated  when  the  rectangle  moves  from  the  distance  2  feet 
to  the  distance  4  feet  ? 

6.  On  the  double  ordinates  of  the  ellipse  h'^x^  +  cfiy^  =  a^b^,  and  in  planes 
perpendicular  to  that  of  the  ellipse,  isosceles  triangles  having  vertical  angles 
2  a  are  erected.     Find  the  volume  of  the  surface  thus  generated. 

7.  A  circle  of  radius  a  moves  with  its  centre  on  the  circumference  of  an 
equal  circle,  and  keeps  parallel  to  a  given  plane  which  is  perpendicular  to  the 
plane  of  the  given  circle  :  find  the  volume  of  the  solid  thus  generated. 

8.  Two  cylinders  of  equal  altitude  h  have  a  circle  of  radius  a  for  their 
common  upper  base.  Their  lower  bases  are  tangent  to  each  other.  Find  the 
volume  common  to  the  two  cylinders. 

136.  Areas:  polar  coordinates.  Suppose  there  is  required  the 
area  of  the  figure  bounded  by  the  curve  whose  equation  is 
/(r,^)  =  0,  and  the  radii  vectores  drawn  to  two  assigned  points 

on  this  curve. 


Q{r2,dii) 


Let  LG  he  the  curve 
f(r,  6)  =  0,  and  F  and 
Q  the  points  (r^,  6i) 
and  (n,  O2)  respectively; 
it  is  required  to  find 
the  area  POQ.  Sup- 
pose that  the  angle  POQ 
is  divided  into  n  equal 
angles  each  equal  to  AO, 
and  let  VOW  be  one  of 
these  angles.  Denote  Fas 
the  point  (r,  0).  Through 
V,   about  0   as  a  centre, 


draw  a  circular  arc  intersecting  0  TF  in  M. 


136.]  AREAS:   POLAR   COORDINATES.  243 

Through  W,  about  0  as  a  centre,  draw  a  circular  arc  intersecting 
OV  in  W.     Denote  MW  hy  A?-. 

Then,  area  OF3/=i?-A<9  (PL  Trig.,  p.  175),  and  area  ONW 
=  i(r  +  Ar)-A^. 

Let  "inner  "  and  "outer"  circular  sectors,  like  VOM and  NOW 
in  the  case  of  VW,  be  formed  for  each  of  the  arcs  like  FTT  which 
are  subtended  by  angles  equal  to  A^  and  lie  between  P  and  Q.  It 
is  evident  that 

total  area  of  inner  sectors  <area  POQ< total  area  of  outer  sectors. 

(1) 

In  the  case  of  the  arc  VW  the  difference  between  the  inner  and 
outer  sectors  is  VMWN.  On  noting  this  difference  for  each  arc 
and  transferring  it  to  the  radius  vector  OPS,  as  indicated  in  the 
figure,  it  is  apparent  that  the  total  difference  between  the  areas 
of  the  inner  and  outer  sectors  is  PBCS.     Now 

area  PBCS  =  area  OSC  -  area  OPB  =  ^{08'-  OP')  AO ; 

and  this  approaches  zero  when  A^  approaches  zero. 

From  these  facts  and  relation  (1)  it  follows  that 

Area  POQ  =  limit  of  area  of  inner  sectors  (or  outer  sectors) 
when  A^  approaches  zero,  that  is,  when  the  number  of  these 
sectors  becomes  infinitely  great.     That  is. 

Area  POQ  =  limit  of  sum  of  areas  of  sectors  VOM  from   OP 
to  OQ  when  A^  approaches  zero 

=  \im:,e=oXi  '""^^  =  I  f  '^r'-dQ.  (See  Art.  96.) 

Note  1.  The  element  of  area  in  polar  coordinates  is  thus  ^r^dd;  this  is 
the  area  of  an  infinitesimal  circular  sector,  of  which  the  radius  is  ?•  and  the 
angle  is  an  infinitesimal,  dd.  The  differential  of  the  area  also  has  the  same 
form  ^  r'^dO.  In  the  element  of  area  dd  must  be  infinitesimal,  in  the  differen- 
tial d9  need  not  be  infinitesimal.     (See  Art.  67  b.) 

Note  2.  It  is  not  necessary  that  the  angles  A9  be  all  equal.  (See  Art.  96, 
Note  3.) 


244  INFI^'UESIMAL    CALCULUS.  [Ch.  XVI. 

1  EXAMPLES. 

J 

1.  Find  the  area  of  a  loop  of  the  curve  r  =  «  sin  2  ^. 

It  is  first  necessary  to  find  tlie  values  of  d  at  the  beginning  and  at  the  end 
of  a  loop.     At  0  (see  Fig.,  page  414)  r  =  0;  hence,  sin2^z=0  at  0.     If 

sin  2  ^  =  0,    then   2^  =  0,    tt,    2  tt,   •••,   and,    accordingly,   ^  =  0,   ~,  tt,   •••. 

Any  pair  of  consecutive  values,  say  0  and  -,  are  values  of  0  at  0  at  the 
beginning  and  end  of  a  loop. 

.-.  area  of  a  loop  =  \(^r'^dd  =  —  Psin2  2  ^  =  -  P  ^  (1  -  cos  4  d)dd 

4  L  4     Jo       « 

2.  Find  the  area  of  one  of  the  loops  of  the  curve  r  =  a  sin  3  6. 

3.  Find  (1)  the  area  of  a  loop  of  the  lemniscate  r^  =  aP-  cos  2  ^  ;  (2)  the 
area  of  a  loop  of  the  curve  r^  =  a^  cos  Jid. 

4.  Show  that  (1)  the  area  included  between  the  hyperbolic  spiral  rd  =  a 
and  any  two  radii  vectores  is  proportional  to  the  difference  between  the 
lengths  of  these  radii  vectores  ;  (2)  the  area  included  between  the  logarithmic 
spiral  r  =  e«^  and  any  two  radii  vectores  is  proportional  to  the  difference 
between  the  squares  on  these  radii  vectores. 

5.  Find  the  area  enclosed  by  the  cardioid  r^  =  a^  cos  — 

2 

6.  Find  the  area  of  the  oval  r  =  3  +  2  cos  6. 

7.  Compute  the  area  of  the  loop  of  the  folium  of  Descartes  x^  +  y^  =  3  a  xy. 

Suggestion  for  Ex.  7  :  Change  to  polar  coordinates,  and  then  use  the 
substitution  z  =  tan  6. 

Note  3.  On  finding  areas  of  curves  by  double  integration.  For  the  sake 
of  illustration  an  example  will  be  shown  in  which  areas,  in  polar  coordinates, 
are  found  by  double  integration. 

8.  Find  the  area  of  the  circle 
r  =  2a  cos  6. 

Take  any  point  P  in  ODA. 
Let  0P=  r,  angle  AOP=d.  Pro- 
duce OP  a  distance  Arto  Q  ;  revolve 
OPq  through  an  angle  A^.  Then 
PQ  sweeps  over  the  area  PQPS. 

Area  PQPS 
=  ^  OQ^ .  A^  -  I  OP  •  A^ 
Fig.  71.  =  r  -  Ar  -  Ad  +  \  (Ar)2  •  A^. 


137.]  LENGTHS   OF  CURVES.  245 

One  can  proceed  to  find  the  limit  of  the  sum  of  the  areas  like  PQBS  in 
01) A,  in  either  of  the  two  following  ways  (a)  and  (b). 

(a)  Starting  with  PQBS  as  an  element  of  area,  find,  the  area  of  the 
sector  BOC]  then,  using  BOC  as  an  element  of  area,  derive  therefrom 
the  area  of  ODA.     Thus, 

r=OB 

XT^  r  r=2  a  cos  0 

area  BOC=  limA,-^  2^  PQBS  =  j  r  dr  -  M  ; 

area  ODA  =  lim a0^ 2^  BOC    =i     j  rdrde  =  J^- 

0=0 

(b)  Starting  with  PQBS  as  an  element  of  area,  find  the  area  of  the 
circular  strip  GDF;  then  using  GDF  as  an  element  of  area,  derive  there- 
from the  area  of  ODA.     Thus, 

area  GDF  =  \imy9=o  2^         PQBS  =  i  ^  -« ^  r  dd  ■  Ar ; 

area  ODA  =  lim^r-:^ ^  GDF  =  f""  T"'  ''  2^ '  rdd  dr  =  ^. 

.-.  area  of  circle  =  2  area  ODA  =  ira^.  [Ex.  4  (7),  Art.  130.] 

In  this  method  of  computing  areas  the  infinitesimal  element  of  area  is 
thus  rdrdd. 

Note  4,  For  discussions  on  the  sign  to  be  given  to  an  area,  on  the  areas 
of  closed  curves,  and  on  the  area  swept  over  by  a  moving  line,  see  Lamb, 
Calculus,  Arts.  99,  101;  Gibson,  Calculus,  §§128,  129^  Echols,  Calculus, 
Arts.  163,  164. 

137.  Lengths  of  curves:  rectangular  coordinates.  Let  it  be  re- 
quired to  find  the  length  of  an  arc  ^^^  . 
PQ  of  the  curve  whose  equation  is 
y  =f(x),  or  F(x,  y)  =  0.  Let  P,  Q 
be  the  points  (x^,  yi),  {x^,  y^  respec- 
tively, and  denote  the  length  of  PQ 
by  s. 

Suppose  that  chords  like  VW 
are  inscribed  in  the  arc  from  P  to 
Q.  Through  V  draw  VN  parallel 
to  the  .-c-axis,  and  through  W  draw     ^  Fig.  72. 


PUi.i/i) 


246  INFINITESIMAL   CALCULUS.  [Ch.  XVI. 

WN  parallel  to  the  ?/-axis.     Let  V  be  (x,  y)  and  TT  be  (ic  +  ^x, 
y-^Ay).     Then  VJS'=Ax,    WN=Ay,  and 


chord  VW=  V  (Ax)^  +  (Ayf  (1) 

Now  suppose  that  Ax,  and  consequently  A?/,  approach  zero; 
then  the  arc  VW  and  the  chord  VW  both  become  infinitesimal. 
The  smaller  the  chords  VW  from  P  to  Q  are  taken,  the  more 
nearly  will  their  sum  approach  to  the  length  of  the  arc  PQ.  The 
difference  between  their  sum  and  the  length  of  FQ  can  be  made 
as  small  as  one  pleases,  simply  by  decreasing  the  arcs.     Thus : 

s  =  limit  of  sum  of  chords  VW  when  these  chords  become 
infinitesimal  * 


=  nm,..o2Vi+(flJ^^ 

=  ^""^  ^1 -\- l^y  -  dx.    (Definitions,  Arts.  22, 23, 96.)    (4) 
Similarly,  from  form  (3), 


iN 


V,  .'■+(S)  ■*■  <") 

Note  1.  The  quantities  under  the  integration  sign  in  (4)  and  (5)  are  the 
infinitesimal  elements  of  length  in  rectangular  coordinates.  The  differential 
of  the  arc  also  has  the  same  forms  (Art.  67  c) ;  see  Note  1,  Art.  136. 

Note  2.  In  (4)  the  integrand  must  be  expressed  in  terms  of  x  ;  in  (5)  in 
terms  of  y. 

Note  .3.  The  process  of  finding  the  length  of  a  curve  is  often  called  the 
rectification  of  the  curve ;  for  it  is  equivalent  to  getting  a  straight  line  of  the 
same  length  as  the  curve,  t 

*  For  rigorous  proof  of  this,  depending  on  elementary  algebra  and  geom- 
etry, see  Rouch^  et  Comberousse.  Traite  de  Geometric  (1891),  Part  I.,  §  291. 
For  a  proof  of  the  same  principle  and  for  interesting  remarks  on  the  length 
and  rectification  of  a  curve,  see  Echols,  Calculus,  Arts.  165,  172. 

t  The  semi-cubical  parabola  was  the  first  curve  that  was  ever  rectified 
absolutely.  William  Neil  (1637-1670),  a  pupil  of  Wallis  at  Oxford,  found 
the  length  of  any  arc  of  this  curve  in  1657.     This  was  also  accomplished 


137.]  LENGTHS   OF  CURVES.  247 

Note  4.  It  has  been  pointed  out  in  Art.  19,  Ex.  6,  Note,  that  the  differ- 
ence between  an  infinitesimal  arc  and  its  chord  is  an  infinitesimal  of  an  order 
at  least  three  lower.  From  this  and  Theorems  A  and  B,  Art.  21,  it  follows 
that  the  limit  of  the  sum  of  an  infinite  number  of  infinitesimal  arcs  is  the 
same  as  the  limit  of  the  sum  of  the  chords  of  these  arcs. 


^ 


EXAMPLES. 
1.   Find  the  length  of  the  four-cusped  hypocycloid  x^  -{-  y'^  =  a^. 


Length  of  a  quadrant  =  C  "-yjl  +  l^Vdx.  (1) 

1 

On  differentiation,  ^x~^  +  - y~^^  =  0  ;  whence  ^  =  f^y. 
S  S       dx  dx     \xj 

,.  a  quadrant  =  ^■^1  +  4"^=  C^'^^'^=  C^  "='  =  -''■ 

.-.  length  of  hypocycloid  =  4xfa  =  6a. 

Note  5.  The  hypocycloid,  sometimes  called  the  astroid,  may  also  be 
represented  by  the  equations  x  =  a  cos^  e,  y  =  a  sin^  6.  (This  may  be  veri- 
fied by  substitution.)     On  using  these  equations  it  follows  that 


dx=  —  Sa  cos2 e sin  Odd,  dy  =  Sa sin2 d cos d dd, 

dy 
dx 


whence  -^  =  —  tan 


Thence  (1)  becomes : 


length  of  quadrant  =  —  i    ^  v  1  +  tan^ 6  -Za cos^ e sin 6 dd 


a  i  "  si 
Jo 


sin  ecosddd  =  —,  as  before. 


(Ex.  Show  that  the  area  of  the  hypocycloid  x  =  a  cos^  0,  y  =  a  sin^  0 
is  I  ircfi  ;  and  that  the  volume  generated  by  its  revolution  about  the  x-axis  is 
T^  wa^,  as  obtained  otherwise  in  Art.  112,  Ex.  20.) 

2.  Find  the  lengths  of  the  following : 

(1)  The  circle  x'^  +  y'^  —  a'^.  (2)  The  arc  of  the  parabola  y'^  =  4  ax,  (a)  from 
the  vertex  to  the  point  {xi,  y\) ;   (&)  from  the  vertex  to  the  end  of  the  latus 

independently  by  Heinrich  van  Heuraet  in  Holland.  The  second  curve  to 
be  rectified  was  the  cycloid.  This  was  effected  by  the  famous  architect, 
Sir  Christopher  Wren  (1632-1723),  in  1673,  and  also  by  the  French  mathe- 
matician, Pierre  de  Fermat  (1601-1665). 


248 


INFINITESIMAL   CALC UL US . 


[Ch.  XVI. 


rectum.     (3)  (a)  The  arc  of  the  cycloid  x  =  a(6  —  sin  6),  y  =  a(l  —  cos  9) 
from  d  =  do  to  e  =  6i;  (b)  a,  complete  arch  of  this  cycloid.     (4)  The  arc  of 

I  X 

the  catenary  ?/  =  ?(e«  +  e  «),  (a)  from  the  vertex  to  (xi,  ?/i);  (b)  from  the 
vertex  to  the  point  for  which  x  =  a. 


,    2 


1.     Thence 


3.  Find   the    whole    length    of    the    curve 
deduce  the  length  of  the  hypocycloid. 

4.  Show  that  in  the  ellipse  x  =  a  sin  0,   y  =  b  cos  0,  0  being  the  com- 
plement of  the  eccentric  angle,  the  arc  s  measured  from  the  extremity  of  the 

minor  axis  is  a  j  Vl  —  e^  sin"-^  <p,  e  being  the  eccentricity.  (This  integral  is 
called  "the  elliptic  integral  of  the  second  kind. " )  Then  show  that  the  perim- 
eter of  an  ellipse  of  small  eccentricity  e  is  approximately  2  7ra[  1  —  —  j. 

138.   Lengths  of  curves :  polar  coordinates.     Let  it  be  required  to 

find  the  length  of  an 
arc  PQ  of  the  curve 
/(r,  e)  =  0.  Let  F  and 
Q  be  the  points  (r^,  O^), 
(vo,  O2),  respectively,  and 
denote  the  length  of  arc 
FQ  by  s.  Suppose  that 
chords  like  VW  are  in- 
scribed in  the  arc  from 
F  to  Q.  Let  Fand  W 
be  denoted  as  the  points 
(r,  0),  (r  +  Ar,  0  +  AO), 
respectively.     Then,  from  Eq.  (2)  Art.  67  d, 


Q(r.,,e^) 


V{r,0) 


chord  VW-. 


r 2 .  sm  1 A^  H •  A^.   (1) 

^M         '        A^y         ^  ^ 


The  length  of  the  arc  FQ  (see  Art.  137)  is  the  limit  of  the  sum 
of  the  lengths  of  the  chords  Fir  from  Pto  Q,  when  these  chords 
become  infinitesimal,  that  is  when  A^  approaches  zero.  Hence, 
from  (1)  and  the  definitions  of  a  derivative  and  an  integral, 


-j>-(ir-- 


(2) 


138,180.]  AREAS   OF  SURFACES.  249 

It  can  also  be  shown  [see  the  derivation  of  result  (6),  Art.  67  d], 

that  s  =  j'7V^2^f,)'  +  l  •  ^^-  (3) 

Note  1.  The  quantities  under  the  integration  sign  in  (2)  and  (3)  are  the 
infinitesimal  elements  of  length  in  polar  coordinates.  The  differential  of  the 
arc  also  has  the  same  forms,  Art.  67  d  ;  see  Note  1,  Art.  137. 

Note  2.  In  (2)  the  integrand  must  be  expressed  in  terms  of  ^  ;  in  (3), 
in  terms  of  r. 

Note  3.    The  intriusic  equation  of  a  curve.     See  Appendix,  Note  B. 

EXAMPLES. 

1.    Find  the  length  of  the  card io Id  r  =  a(l  —  cos  d). 


.  =  2£7^V.(|)^.. 


The  substitution  of  the  value  of  r  and  --  in  the  integrand  and  simplifica- 
^.  .  d9 

tion,  give 


=  2aV2  f^Vl  -coHdde  =  4a  (''  siu^  dd  =  Sa. 
Jo  Jo  2 


2.    Find  the  lengths  of  the  following  : 

(1)  The   circle  r  =  a.       (2)  The   circle   r  r=  2  a  sin  ^.        (3)  The   curve 

a 

7'  =  asm^~-     (4)  The  arc  of  the  equiangular  spiral  r  =  «e^«^ta,   (a)  from 

o 

^  =  0  to  ^  =  2  TT  ;  {b)  from  ^  =  2  tt  to  ^  =  4  tt.     (5)  The  arc  of  the  spiral  of 
Archimedes  r  =  ad  from  (ri,  ^i)  to  (r2,  62) •     (6)  The  arc  of  the  parabola 

r  =  a  sec2  -,  (a)  from  6*  =  0  to  ^  =  ^1;  (6)  from  e  =  --tod  =  +  -• 

139.   Areas  of  surfaces  of  revolution. 

Note  1.  Geometrical  Theorem.  Let  KL  and  BS  (Fig.  74  a)  be  in  the 
same  plane.  In  elementary  solid  geometry  it  is  shown  that  if  a  finite  straight 
Ihie  KL  makes  a  complete  revolution  about  RS,  the  surface  thus  generated  by 
KL  is  equal  to  2wT3I  •  KL,  in  which  TM  is  the  length  of  the  perpendicular 
lat  fall  on  RS  from  T,  the  middle  point  of  KL. 

Suppose  that  an  arc  PQ  of  a  curve  y  =f(x)  revolves  about  the 
a'-axis,  and  that  the  area  of  the  surface  thus  generated  is  required. 


250 


INFINITESIMAL   CALCULUS. 


[Ch.  XVI. 


Let  P  and  Q  be  the  points  (xi,  ?/i)  and  (xg,  2/2)  respectively.  Sup- 
pose that  PQ  is  divided  into  small  arcs  such  as  KL,  and  denote 
K  and  L  as  the  points  (x,  y)  and  {x  -f-  Ao?,  ?/  +  ^V)  respectively. 


QkXo.y^ 


M 

Fig.  74  a. 


SO 


M 
Fig.  74  6. 


Draw  the  chord  KL,  and  from  T,  the  middle  point  of  this  chord, 
draw  TM  at  right  angles  to  the  ic-axis.  Then  the  area  generated 
by  the  chord  KL  when  the  arc  PQ  revolves  about  the  x-axis 

=  27rTM-KL 


=  2  TT (2/  +  i  Ay)yjl  +  ff)  •  Ax.       (Note  1.) 


\Ax 


The  smaller  the  chords  KL  are  taken,  the  more  nearly  will  the 
surfaces  generated  by  them  approach  coincidence  with  the  surface 
generated  by  the  arc  PQ,  and  the  difference  between  area  of  the 
latter  surface  and  the  sum  of  the  areas  of  the  former  surfaces 
can  be  made  as  small  as  one  pleases  by  decreasing  Ax.  Accord- 
ingly, the  area  of  the  surface  generated  by  the  arc  PQ  is  the 
limiting  value  of  the  sum  of  the  areas  of  the  surfaces  generated 
by  the  chords  KL  (from  P  to  Q)  when  these  chords  become 
infinitesimal.     That  is,  area  of  surface  generated  by  JPQ 


=  liin^,^„52  ^(y  +  |  ^y)^l  +  (|^)'a» 


(1) 


«     r*2       L  ,  fdv\2^        (Definitions  of  derivative    .t.. 
2-lj  H^  +  [^^)  ^^'        and  integral.)  (2) 


139.] 


AREAS   OF  SURFACES. 


251 


If  the  length  of  the  chord  KL  be  denoted  by  Jl  +  f—\^y, 
this  integral  takes  the  form  \^^/ 


surface 


=2.fyyji+(^Ji)W 


dy 


(3) 


Note  2.  Each  of  the  expressions  to  be  integrated  in  (2)  and  (3)  may  be 
denoted  by  2  iry  ds  [Art.  67  /(9)],  and  is  called  an  element  of  the  surface 
of  revolution. 

If  PQ  is  revolved  about  the  y-axis,  the  element  of  surface  is  2  ira?  ds ; 
and  the  surface 


dy 


dy 


(4) 


The  questions,  whether  to  use  form  (2)  or  (3),  and  which  of  (4)  to  employ, 
are  decided  by  convenience  and  ease  of  working.  (See  Art.  136,  Note  1,  and 
Art.  67/0 

Note  3.  In  a  similar  manner  it  can  be  shown  that  the  area  of  the  surface 
generated  by  the  revolution  of  an  arc  of  a  curve  about  any  straight  line  in 
the  plane  of  the  arc,  is  ^^ 

2  7ri    \ds,  (5) 

in  which  ds  denotes  an  infinitesimal  arc  of  the  curve,  I  the  distance  of  this 
infinitesimal  arc  from  the  straight  line,  and  e\  and  e^  are  coordinates  of  some 
kind  that  denote  the  ends  of  the  revolving  arc.  An  illustration  is  given  in 
Ex.  4. 

EXAMPLES. 


1.    Find  the  surface  generated  by  the  revolution  of   the   hypocycloid 
a;3  +  ?/3  =  ^3  about  the  a;-axis. 
Surface 
=  2i'^''2Tr'PN'ds 

Jx=0 


Jo  ^  ^         . 


(See  Art.  137,  Ex.  1.) 
6  Tr«3  p(al  _  x3)^d(a^  -  x^) 


2.    X 


=  \^ra^ 


Fig.  75. 


252 


INFINITESIMAL    CA L C UL  ITS. 


[Ch.  XVI. 


V- 


In  this  case  an  easier  integral  is  obtained  by  expressing  the  surface  in 
terms  of  y  and  dy^  as  in  form  (3) .     Thus, 

Surface  =  2-2  tt  P"^"?/^!  -i-l^y^Ydy  =  4  iraH^jKly  =  V  -jra^, 

2.  Calculate  the  surface  of  the  hypocycloid  in  Ex.  1,  using  the  equations 
X  =  a  cos^  d,  y  =  a  sin^  6. 

3.  Derive  formula  (5). 

4.  The  cardioid  r  =  «(1  —  cos^)  revolves  about  the  initial  line  :  find  the 
area  of  the  surface  generated. 

Surface  =  2  tt  i        PN  •  ds. 

Je=o  _    

Now   P^=  rsin^  =  a(l  —  cos^)sin  ^,    and  ds  =  aV2Vl  —  cos d dd  (see 
Ex.  1,  Art.  138). 


(9=7r 


6>-=0 


.*.  surface  =  2-N/2  7ra2  \^  {I 


Fig.  76. 


cos^)"2  sin  Odd 


rV2 

L    5 


7r«2(l 


5"!' 


5.  Find  the  area  of  the  spherical  surface  generated  by  the  revolution  of  a 
circle  of  radius  a  about  a  diameter. 

6.  A  quadrant  of  a  circle  of  radius  a  revolves  about  the  tangent  at  one 
extremity.     What  is  the  area  of  the  curved  surface  generated  ? 

7.  Calculate  the  area  of  the  surface  of  the  prolate  spheroid  generated  by 
the  revolution  of  the  ellipse  b'^x^  4-  a^y^  —  a^b"^  about  the  a:-axis. 

8.  In  the  case  of  an  arch  of  the  cycloid  x  =  a(d  —  sm6),  y  =  a(l—cos6), 
compute  :  (1)  the  area  between  the  cycloid  and  the  cc-axis ;  (2)  the  volume 
and  the  surface  generated  by  its  revolution  about  the  x-axis  ;  (3)  the  volume 
and  the  surface  generated  by  its  revolution  about  the  tangent  at  the  vertex. 

9.  Find  the  volume  and  the  surface  generated  by  revolving  the  circle 
x^  j-{y  —  by  =  a'^  (&  >  a),  about  the  x-axis. 


140.]  AREAS   OF  SURFACES.  253 

10.  Find  the  area  of  the  surface  generated  by  the  revolution  of  the  arc 
of  the  catenary  in  Ex.  6,  Art.  112. 

11.  Tlie   arc   of  the   curve    7-  =  a  sin  2  6,    from    ^  =  0   to   d  =-  (<.e.  the 

4 
first  half  of  the  loop  in  the  first  quadrant),  revolves  about  the  initial  line  : 
find  the  area  of  the  surface  generated.     What  is  the  area  of  the  surface 
generated  by  the  revolution  of  the  second  half  of  the  same  loop  about  the 
same  line  ? 

12.  A  circle  is  circumscribed  about  a  square  whose  side  is  a.  The  smaller 
segment  between  the  circle  and  one  side  of  the  square  is  revolved  about 
the  opposite  side  of  the  square.  Find  the  volume  and  the  surface  of  the 
solid  ring  thus  generated. 

140.  Areas  of  surfaces  whose  equations  have  the  form  z  =/(«,  y) 

or  F{pc,  y,  z)  =0.     It  is  shown  in  solid  geometry  that: 

(a)  The  cosine  of  the  angle  between  the  ac«/-plane  and  the  tangent  plane 
at  any  point  (.r,  ?/,  z)  on  such  a  surface,  supposed  to  be  continuous,  is 

{-(l)^-(l)f- 

(&)  The  area  of  the  projection  of  a  segment  of  a  plane  upon  a  second 
plane  is  obtained  by  multiplying  the  area  of  the  segment  by  the  cosine  of 
the  angle  between  the  planes. 

It  follows  from  (a)  and  (6)  that : 

(c)  If  there  be  an  area  on  the  a;?/-plane  equal  to  A,  then  A  is  the  area 
that  would  be  projected  on  the  a;2/-plane  by  an  area  on  the  tangent  plane  at 
(a;,  ?/,  z)  which  is  equal  to 

(See  C.  Smith,  Solid  Geometry,  Arts.  206,  20,  31 ;  Murray,  Integral  Calcu- 
lus, Art.  75.) 

Let  z  =f(x,  y)  be  the  equation  of  a  surface  BFCRAGB  [Fig.  66]  whose 
area  is  required.  Let  P(x,  y,  z)  be  any  point  on  this  surface,  and  Pi  the 
point  (cc,  y,  0)  vertically  below  P.  Let  P\Q\  be  a  rectangle  in  the  x2/-plane 
having  its  sides  equal  to  Ax  and  Ay  respectively,  and  parallel  to  the  x-  and 
?/-axes.  Through  the  sides  of  this  rectangle  pass  planes  perpendicular  to  the 
x^-plane,  and  let  these  planes  make  with  the  surface  the  section  PQ,  and 
with  the  tangent  plane  at  P  the  section  PQ^-  {Q\Q  produced  is  supposed 
to  meet  in  Q2  the  tangent  plane  at  P.) 

Then,  area  Pi  ^1  =  Ax  •  Ay. 


Hence,  by  (2),        area  PQ.,  =  a^I  +  (—Y-^  ifY'^^  '  ^' 


254  INFINITESIMAL   CALCULUS.  [Ch.  XVI. 

Now  the  smaller  Ax  and  Ay  become,  the  more  nearly  will  the  section  PQ2 
on  the  tangent  plane  at  P  coincide  with  the  section  PQ  on  the  surface. 
Accordingly,  the  more  nearly  will  the  sum  of  the  areas  of  sections  like  PQ2 
on  the  tangent  planes  at  points  taken  close  together  on  the  surface,  become 
equal  to  the  area  of  the  surface  ;  moreover,  the  difference  between  this  sum 
and  the  area  of  the  surface  can  be  made  as  small  as  one  pleases.  Con- 
sequently, the  area  of  the  surface  is  the  limit  of  the  sum  of  the  areas  of 
these  sections  on  the  tangent  planes  when  these  sections  become  infinitesimal. 

That  is,  

ave^BFCRAGB=(''-''^  P^"^  Jl  +  f/^V+ fi^^.  dt/c?^. 

J.e=0         J!/=0         ^  \dOCJ  \dy) 

Note.     The  integral  f  y~^^^]l  +  (— y+  (^Ydyldx  gives  the  area 

rx=OA 
of  the  strip  or  zone  POL,  and  the  integral  \         BGLdx  gives  the  sum  of 

these  zones  from  BOG  to  A. 

EXAMPLES. 

1.  Find  the  area  of  the  portion  of  the  surface  of  the  sphere  in  Ex.  7, 
Art.  132,  that  is  intercepted  by  the  cylinder. 

The  area  required  =  4  area  AVBLA  (Fig.  68).    In  this  figure,  the  equation 

of  the  sphere  is  x^  +  y^  -\-  z^  =  a^, 

and  the  equation  of  the  cylinder  is  x"^  +  y^  =  ax. 

The  area  of  a  strip  L  V,  two  of  whose  sides  are  parallel  to  the  zy-plane,  will 
first  be  found ;  then  the  sum  of  all  such  strips  in  the  spherical  surface 
AVBLA  will  be  determined. 

Since  the  required  surface  is  on  the  sphere,  the  partial  derivatives  must  be 
derived  from  the  equation  of  the  sphere. 

Accordingly,  ^  =  -?,    ^  =  -??; 

hence,  1  +  f^V+  (^^V=  1  +^  +  ^  =  ^^^ 

\dx)  \dy)  Z'        Z'^        Z^         a2  _  a;2  _  y2 


Also,  BK  =  Vax 


.'.  area  AVBLA  =     \     \  "        -  dy  dx 


]Vax-xi 

Va2  -  X2- 


a  \       sin-i —    y  dx 


•r 


V=i 


dx. 


# 


141.]  MEAN   VALUES.  255 

This  integral  can  be  evaluated  by  integrating  by  parts.  The  integration 
can  be  simplified  by  means  of  the  substitution  sin  z  =\, — - —  It  will  be 
found  that  area  required  =  4  area  AVE  LA  =  2  (tt  -  2)d^  =  2.2832  a^. 

2.  Find  the  area  of  the  surface  of  the  cylinder  intercepted  by  the  sphere 
in  Ex.  7,  Art.  132. 

3.  By  the  method  of  this  article,  find  the  surface  of  the  sphere  x-  +  y'^ 

+  Z'^  —  «2. 

4.  A  square  hole  is  cut  through  a  sphere  of  radius  a,  the  axis  of  the 
hole  coinciding  with  a  diameter  of  the  sphere  :  find  the  volume  removed  and 
the  area  of  the  surface  cut  out,  the  side  of  a  cross-section  of  the  hole  being  2  h. 

5.  Find  the  area  of  that  portion  of  the  surface  of  the  sphere  inter- 
cepted by  the  cylinder  in  Ex.  4,  Art.  133. 

141.  Mean  values.  In  Art.  98  it  has  been  stated  that  if  the 
curve  y=zf{x)  be  drawn  (Fig.  44),  and  if  0^1  =  a  and  OB  =  h, 
then,  of  all  the  ordinates  from  A  to  B, 

the  mean  value  =  '^''^'^  f^^^  =  jji^^ 

AB  b-a  ^  ^ 

Result  (1)  can  be  derived  in  the  following  way  which  has 
also  the  advantage  of  being  adapted  for  leading  up  to  a  more 
general  notion  of  mean  value.  The  mean  value  of  a  set  of  quan- 
tities is  defined  as 

the  sum  of  the  values  of  the  quantities 
the  number  of  the  quantities 

For  instance,  if  a  variable  quantity  takes  the  values  2,  5,  7,  9, 
the  mean  of  these  values  is  ^^- or  54. 


4 


Now  take  any  variable,  say  x,  and  suppose  that  f(x)  is  a  con- 
tinuous function,  and  let  the  interval  from  x  =  a  to  x  =  b  be 
divided  into  n  parts  each  equal  to  Ax,  so  that  n  Ax  =  b  —  a.  Let 
the  mean  of  the  values  of  the  function  for  the  n  successive  values 

'  a,  a-{-  Ax,  a +  2  Ax,  •  ••,  a-\-n  —  l  Ax, 

be  required.     The  corresponding  n  successive  values  of  the  func- 
lon  are    ^^^^^  ^^^^  _^  ^^^^  ^^^  _^  ^  ^^^^^       ^  ^^^  _^  —-j  ^  ^_^^^ 


256 


INFINITESIMAL   CALCULUS. 


[Ch.  XVI. 


Hence,  mean  value  of  function 


^  /(^)  +/(^  +  Ao;)  +/(a  +  2  Aa^)  +  ♦  ■ .  +/(«  +  ii  -  1  •  Aa;) 


(2) 


Now  n  ^x  =  h  —  a,  whence  n  = Substitution  in  (2)  gives 


mean  value 


Ax 


^  f(a)Ax  -\-f(a  +  Aa;)  Ax +/(a  +  2  Ax)  Aa?  H \-f(a  +  n—l  Aa;^  A  g  ^ 

6 -a  '  ^3* 

Finally,  let  the  mean  of  all  the  values  that  f(x)  takes  as  x  varies 
from  a  to  6  be  required.  In  this  case  n  becomes  infinitely  great 
and  Ax  becomes   infinitesimal ;    accordingly   [Art.   96   (2),   (3)], 


(3)  becomes 


mean  value 


fV(x) 

Ja 


dx 


h  —  a 


(4) 


as  already  represented  geometrically  in  Art.  98. 


Note  1.    Reference  for  collateral  reading.     Ecliols,  Calculus,  Arts. 
150-152. 

EXAMPLES. 

1.    Find  the  mean  length  of  the  ordinates  of  a  semicircle    (radius  a). 
the  ordinates  being  erected  at  equidistant  intervals  on  the  diameter. 

Choose  the  axes  as  in  Fig.  77.  Then  the  equation  of  the  circle  is 
x-^  +  2/^  =  «'^.  Let  PN  denote  any  of  the  ordi- 
nates drawn  as  directed. 


Mean  value 


■  P"  PN .  dx      ("  Va'-  -  y^^  dx 


(-«) 


2a 


ira^ 


2.2a 


,7854  a. 


2.  Find  the  mean  length  of  the  ordinates  of  a 
semicircle  (radius  a),  the  ordinates  being  drawn  at 
equidistant  intervals  on  the  arc. 

Let  PN  be  any  of  the  ordinates  drawn  at  equi- 
distant intervals  on  the  arc,  that  is,  at  equal  incre- 
ments of  the  angle  6. 


Mean  value  =  '^^^ 


i 


e=w 


PN'de 


y    a  sin 


Odd 


=  ^JL  =  .Q^ma. 


141.]  MEAN   VALUES.  257 

Note  2.  A  slight  inspection  will  show  that  it  is  reasonable  to  expect  the 
results  in  Exs.  1,2,  to  differ  from  each  other. 

Suggestion  :  Draw  a  number  of  ordiiiates,  say  4  or  6  or  8,  as  specified 
in  Ex,  1,  and  compare  them  with  the  ordinates  of  equal  number  drawn  as 
specified  in  Ex.  2. 

3.  Find  the  average  value  of  the  following  functions:  (1)  1  x--\-^x  —  ^ 
as  X  varies  continuously  from  2  to  6  ;  (2)  x-^  —  3  x^  -f  4  a;  +  11  as  x  varies  from 
—  2  to  3.     Draw  graphs  of  these  functions. 

4.  Find  the  average  length  of  the  ordinates  to  the  parabola  y'^  =  ^x 
erected  at  equidistant  intervals  from  the  vertex  to  the  line  x  =  6. 

5.  (1)  In  Fig.  51  find  the  mean  length  of  the  ordinates  drawn  from 
OiV  to  the  arc  OML,  and  the  mean  length  of  the  ordinates  drawn  from  OiVto 
the  arc  ORL.  (2)  In  Fig.  50  find  the  mean  length  of  the  abscissas  drawn 
from  OY,  (a)  to  the  arc  OR;  (b)  to  the  arc  RL;  (c)  to  the  arc  ORL. 
(3)  In  Fig.  52  find  the  mean  ordinate  from  OL,  (a)  to  the  arc  TKN;  (&)  to 
the  arc  TGM. 

6.  (1)  In  the  ellipse  whose  semiaxes  are  6  and  10,  chords  parallel  to 
the  minor  axis  are  drawn  at  equidistant  intervals :  find  their  mean  length. 
(2)  In  the  ellipse  in  (1)  find  the  mean  length  of  the  equidistant  chords  that 
are  parallel  to  the  major  axis.  (3)  Do  as  in  (1)  and  (2)  for  the  general  case 
in  which  the  major  and  minor  axes  are  respectively  2  a  and  2  6. 

7.  On  the  ellipse  in  Ex.  6,  (3),  successive  points  are  taken  whose  eccen- 
tric angles  differ  by  equal  amounts :  find  the  mean  length  of  the  perpen- 
diculars from  these  points,  (1)  to  the  major  axis  ;  (2)  to  the  minor  axis. 

8.  In  the  case  of  a  body  falling  vertically  from  rest,  show  that  (1)  the 
mean  of  the  velocities  at  the  ends  of  successive  equal  intervals  of  time,  is  one- 
half  the  final  velocity  ;  (2)  the  mean  of  the  velocities  at  the  ends  of  succes- 
sive intervals  of  space,  is  two-thirds  the  final  velocity.  (The  velocity  at  the 
end  of  t  seconds  is  ^  gt  feet  per  second  ;  the  velocity  after  falling  a  distance 
s  feet  is  V2  gs  feet  per  second.) 

9.  A  number  n  is  divided  at  random  into  two  parts  :  find  the  mean  value 
of  their  product. 

10.  Find  the  mean  distance  of  the  points  on  a  circle  of  radius  a  from 
a  fixed  point  on  the  circle. 

The  interval  &  —  a  in  (1)  and  (4)  through  which  the  variable  x 
passes  is  called  the  range  of  the  variable,  and  dx  is  an  infini- 
tesimal element  of  the  range.  In  (1)  and  Ex.  1  the  range  is  a 
particular  interval  on  the  aj-axis.  In  Ex.  2  the  range  is  a  certain 
angle,  namely  tt  ;  in  Ex.  8  (2)  the  range  is  a  vertical  distance  j  in 


258  INFINITESIMAL    CALCULUS.  [Ch.  XVI. 

Ex.  8  (1)  the  range  is  an  interval  of  time.  There  are  various 
other  ranges  at  (or  for)  whose  component  parts  a  function  may 
take  different  values.  For  instance,  a  curved  line  as  in  Ex.  10,  a 
plane  area  as  in  Exs.  11,  13 ;  a  curved  surface  as  in  Ex.  15  (1) ;  a 
solid  as  in  Exs.  16,  17.  The  definition  of  mean  value  [or  result 
(4)]  may  be  extended  to  include  such  cases,  thus : 

lim  2  {(value  of  function  at  each  infini- 
tesimal element  of  the  range)  x  (this 

the  mean  value  of  a  fiinc- )  _     infinitesimal  element)} ^ 

tion  over  a  certain  range  >  the  range 

11.    Find  the  mean  square  of  the  distance  of   a  point  within  a  square 
(side  =  a)  from  a  corner  of  the  square. 

In  tliis  case  "the  range"  extends  over  a  square. 
Choose  the  axes  as  shown  in  Fig.  79.  Take  any  point 
P  (x,  y)  in  tlie  range,  and  let  its  distance  from  O  be 
d.  At  P  let  an  infinitesimal  element  of  the  range 
be  taken,  viz.  an  element  in  the  shape  of  a  rectangle 
whose  area  is  dy  dx.  Now  d^  =  x"^  -{-  y^.  .*.  mean 
value  of  c?2  for  all  points  in 
Fig.  79. 

("  P(a;-  +  2/-)f/2/(^a^ 
OACB 


I 

B 

c 

' ' 

a« 

Ay 

i 

/ 

3 

0 

A      X 

a 

area  of  square 


12.  Find  (1)  the  mean  distance,  and  (2)  the  mean  square  of  the  distance, 
of  a  fixed  point  on  the  circumference  of  a  circle  of  radius  a  from  all  points 
within  the  circle.     (Suggestion  :  use  polar  coordinates.) 

13.  Find  (1)  the  mean  distance,  and  (2)  the  mean  square  of  the  distance, 
of  all  the  points  within  a  circle  of  radius  a  from  the  centre. 

14.  Find  the  mean  latitude  of  all  places  north  of  the  equator. 

15.  For  a  closed  hemispherical  shell  of  radius  a  calculate  (1)  the  mean 
distance  of  the  points  on  the  curved  surface  from  the  plane  surface  ;  (2)  the 
mean  distance  of  the  points  on  the  plane  surface  from  the  curved  surface, 
distances  being  measured  along  lines  perpendicular  to  the  plane  surface. 

16.  Calculate  (1)  the  mean  distance,  and  (2)  the  mean  square  of  the  dis- 
tance, of  all  points  within  a  sphere  of  radius  a,  from  a  fixed  point  on  the 
surface. 

17.  Calculate  (1)  the  mean  distance,  and  (2)  the  mean  square  of  the  dis- 
tance, of  all  points  within  a  sphere  of  radius  a,  from  the  centre. 


141.]  MEAN    VALUES.  259 

18.  Find  (1)  the  mean  distance,  and  (2)  the  mean  square  of  the  distance, 
of  all  points  on  the  surface  of  a  sphere  of  radius  a,  from  a  fixed  point  on  the 
surface. 

19.  Find  (1)  the  mean  distance,  and  (2)  the  mean  square  of  the  distance, 
of  all  points  on  a  semi-undulation  of  the  sine  curve  y  =  asin  x,  from  the 
X-axis. 

Note  3.  The  square  root  of  the  mean  square  in  Ex.  19  (2)  (viz.  .7071  a) 
is  of.  special  importance  in  the  measurement  of  alternating  currents  ;  for  the 
heating  and  dynamometer  effects  of  any  current  depend  directly  upon  this 
square  root.  The  latter  is  generally  called  "  the  mean  square  value  of  the 
ordinate  of  the  sine  curve"  to  distinguish  it  from  "the  average  value"  of  this 
ordinate  as  found  in  Ex.  19  (1). 


CHAPTER   XVII. 

CONCAVITY  AND  CONVEXITY.    CONTACT  AND  CURVA- 
TURE.    EVOLUTES  AND  INVOLUTES. 

142.   Concavity  and  convexity  of  curves  :  rectangular  coordinates. 

Definition.  At  a  point  on  a  curve  the  curve  is  said  to  be  con- 
cave to  a  line  {or  to  a  point  off  the  curve)  when  an  infinitesimal  arc 
containing  the  point  lies  between  the  tangent  at  the  point  and  the 
given  line  (or  point  off  the  curve).  If  the  tangent  lies  between 
the  line  (or  point)  and  the  infinitesimal  arc,  the  arc  there  is  said 
to  be  convex  to  the  line  (or  point). 

Thus,  in  Fig.  20  a,  at  P  the  curve  J/iVis  concave  to  the  line  OX,  and  con- 
cave to  the  point  A  ;  in  Fig.  20  &,  at  Pi  the  curve  MN  is  convex  to  the  line 
OX,  and  convex  to  the  point  A.  The  arc  on  one  side  of  a  point  of  inflexion 
is  concave  to  a  given  line  (or  point),  and  the  arc  on  the  other  side  of  the 
point  of  inflexion  is  convex  to  this  line  (or  point)  (see  Figs.  36  a,  b). 

The  curves  passing  through  P  and  R  have  the  concavity  towards 
the  .T-axis,  and  the  curves  passing  through  Q  and  S  are  convex 

to  the  a^axis.      At  P  ?/  is  positive; 

and    -4  is  negative,  for  -^  decreases 
dx-  dx 

as    a   point    moves   along   the    curve 

towards  the  right  through  P.     At  E 

y  is  negative ;    and   -J-    is   positive, 
1  dxr      ^ 

for   —   increases   as   a   point   moves 
dx 

along  the  curve  towards  the  right  through  R.  Hence,  at  points 
where  a  curve  is  concave  to  the  x-axis  y  ^=-^  is  negative.  A  similar 
examination  of  the  curves  passing  through  Q  and  8  shows  that  at 
points  where  a  curve  is  convex  to  the  x-axis  y  y—  is  positive. 

260 


142,  143.] 


CONTACT. 


261 


Ex.  1.    Prove  the  theorem  last  stated. 


Ex.  2.  Test  or  verify  the  above  theorems  and  Note  1  in  the  case  of  a  num- 
ber of  the  curves  in  the  preceding  chapters. 

Note  1.     The  curves  passing  through  P  and  ^S'  are  concave  doimiwards, 

and  here  y^  is  negative.     The  curves  passing  through  B  and  Q  are  concave 

upwards,  and  liere  —^  is  positive. 

Note  2.  A  point  where  a  curve  stops  bending  in  one  direction  and  begins 
to  bend  in  the  opposite  direction  as  at  L,  A,  Z>,  if,  (r,  P,  Figs.  36  a,  6,  37, 
is  called  a  point  of  inflexion. 


Note  3.     A  curve  /(r,  ^)  =  0  is  concave  or  convex  to  the  pole  at  the  point 

,  e)  according  as  «  +  ^  is  positive  or 
dff^ 

McMahon  and  Snyder,  Diff.  Cal,  Art.  144.) 


(r,  0)  according  as  «  +  ^^  is  positive  or  negative,   u  denoting  -.     (See 


143.   Order  of  contact.     If  two   curves,  y  =  <f}(x)  and  y—f(x), 
intersect  at  a  point  at  which  x  =  a,  as  in  Fig.  81  a,  then  <^(a)  =f(a) 

and  <^'(^)  =?^/'(^0-  ^^  *^W  =./'W  ^^^  <^'(^)  =/'(^)j  ^hen  the  curves 
touch  as  in  Fig.  81  b,  and  they  are  said  to  have  contact  of  the  first 
order,  provided  that  <^"(a)  ^f"(a).  If  4>{a)  =f(a),  <^'(a)  =/'(a), 
and  <f>"(a)  =f"{ci),  but  <^"'(a)  =^f"(ci),  then  the  curves  are  said  to 


y-f(x) 


y-f  (X) 


Fig.  81  a 


have  contact  of  the  second  order,  as  in  Fig.  81  c.  And,  in  general, 
if  <^(a)  =/(«.)  and  the  respective  successive  derivatives  of  <^(x) 
and  f(x)  up  to  and  including  the  nth,  but  not  including  the 
(n  -\-  l)th,  are  equal  for  x  =  a,  then  the  curves  are  said  to  have  con- 
tact of  the  nth  order.  Hence,  in  order  to  find  the  order  of  contact 
of  two  curves  compare  the  respective  successive  derivatives  of  y 
for  the  two  curves  at  the  points  through  which  both  curves  pass. 


262  INFINITEmMAL   CALCULUS.  [Ch.  XVII. 

Note  1.  Another  way  of  regarding  contact  is  the  following.  In  analytic 
geometry  the  tangent  at  P  (Fig.  82  a)  is  defined  as  the  limiting  position 
which  the  secant  PQ  takes  when  PQ  revolves  about  P  until  the  point  of 
intersection  Q  coincides  with  P.  Tlie  line  then  has  contact  of  the  first  order 
with  the  curve.  This  notion  of  points  of  intersection  of  a  line  and  a  curve 
becoming  coincident  will  now  be  extended  to  curves  in  general.    Two  curves, 


C, 
Fig.  82  a.  Fig.  82  h. 

C\  and  C2  (Fig.  82  5),  are  said  to  intersect  when  they  have  a  point,  as  P,  in 
common.  They  are  said  to  have  contact  of  the  first  order  at  P  when  the 
curves  (see  Fig.  82  c)  have  been  modified  in  such  a  way  that  a  second  point 

of  intersection  Q  moves  into  coincidence  with  P.     (The  value  of  -j^  at  P  is 

then  the  same  for  both  curves,  according  to  the  definition  of  a  tangent  as 
given  above.)  The  curves  are  said  to  have  contact  of  the  second  order  at  P 
when  the  curves  have  been  further  modified  in  such  a  way  that  a  third  point 
of  intersection  B  moves  into  coincidence  with  P  and  Q  (see  Fig.  82  d).    (The 


dx  \dx)''  "■""  dx^ 

general,  the  curves  are  said  to  have  contact  of  the  nth  order  at  a  point  P  when 
n  +  1  of  their  points  of  intersection  have  moved  into  coincidence  with  P. 
(At  P  the  respective  derivatives  of  y  up  to  the  nth  are  then  the  same  for  both 
curves.)     See  Echols,  Calculus^  Art.  98. 

Note  2.  In  general  a  straight  line  cannot  have  contact  of  an  order  higher 
than  the  first  with  a  curve.  For  in  order  that  a  line  have  contact  of  the  first 
order  with  a  curve  at  a  given  point,  the  ordinates  of  the  line  and  the  curve 
must  be  equal  there,  and  likewise  their  slopes  ;  thus  two  equations  must  be 
satisfied.  These  equations  suffice  to  determine  the  two  arbitrary  constants 
appearing  in  the  equation  of  a  straight  line.  For  example,  if  the  line 
y  =  mx  +  h  has  contact  of  the  first  order  with  the  curve  y  =  f(x)  at  the  point 
for  which  x  =  a,  the  following  two  equations  are  satisfied,  viz. : 

f(a)  =  ma  +  b,  f'(a)  =  m  ; 

from  these  equations  w  and  h  can  be  found. 

Tliis  line  and  curve  have  contact  of  the  second  order  in  the  particular  (and 
exceptional)  case  in  which  f"{a)  =0j  consequently  (Art.  78),  if  there  is  a 


143.]  CONTACT.  263 

point  of  inflexion  on  the  curve  y  —  /(x)  where  x  =  a,  the  tangent  there  has 
contact  of  the  second  order. 

The  theorem  at  the  beginning  of  this  note  is  also  evident  from  geometrical 
considerations.  Since,  in  general,  a  line  can  be  passed  through  only  two 
arbitrarily  chosen  points  of  a  curve,  it  is  to  be  expected  from  Note  1  that  in 
general  a  line  and  a  curve  can  have  contact  of  the  first  order  only. 

Note  3.  In  general,  a  circle  cannot  have  contact  of  an  order  higher  than 
the  second  with  a  curve.  For  in  order  that  a  circle  have  contact  of  the  second 
order  with  a  curve  at  a  given  point,  three  equations  must  be  satisfied,  and 
these  equations  just  suffice  to  determine  the  three  arbitrary  constants  that 
appear  in  the  general  equation  of  a  circle  [see  Eq.  (2),  Art.  144].  This 
theorem  is  also  evident  from  Note  1  and  the  fact  that,  in  general,  a  circle  can 
be  passed  through  only  three  arbitrarily  chosen  points  of  a  curve.  (In  a  few 
very  special  instances  a  circle  has  contact  of  the  third  order  with  a  curve. 
See  Ex.  4,  Art.  149.) 

Note  4.  It  is  shown  in  Art.  182  that  inhen  two  curves  have  contact  of  an 
odd  order,  they  do  not  cross  each  other  at  the  point  of  contact ;  but  lohen  they 
have  contact  of  an  even  order,  they  do  cross  there.  Illustrations :  the  tangent 
at  an  ordinary  point  on  a  curve,  as  shown  in  Figs.  15,  17  ;  the  tangent  at  a 
point  of  inflexion,  as  in  Figs.  31  a,  h,  36,  37  ;  an  ellipse  and  circles  having 
contact  of  second  order  therewith  (see  Ex.  4,  Art.  149).  This  theorem  may 
also  be  deduced  from  geometry  and  the  definitions  given  in  Note  1. 

N.B.  As  far  as  possible  make  good  figures  showing  the  curves,  lines,  and 
points  mentioned  in  the  exercises  in  this  chapter. 


EXAMPLES. 

1.  Find  the  place  and  order  of  contact  of  (1)  the  curves  y  =  t^  and 
y  =  6  x2  -  9  X  +  4  ;  (2)  the  curves  y  -  x^  and  ?/  =  6  x^  -  12  x  -f  8. 

2.  Determine  the  parabola  which  has  its  axis  parallel  to  the  ?/-axis,  passes 
through  the  point  (0,  3),  and  has  contact  of  the  first  order  with  the  parab- 
ola 2/  =  2  x^  at  the  point  (1,  2). 

3.  What  must  be  the  value  of  a  in  order  that  the  parabola  y  =  x  +  1 
+  a(x— 1)2  may  have  contact  of  the  second  order  with  the  hyperbola 
xy  =  3  X  -  1  ? 

4.  Find  the  parabola  whose  axis  is  parallel  to  the  y-axis,  and  which  has 
contact  of  the  second  order  with  the  cubical  parabola  y  =  x^  at  the  point 
(1,  1). 

5.  Determine  the  parabola  which  has  its  axis  parallel  to  the  y-axis  and  has 
contact  of  the  second  order  with  the  hyperbola  xy  =  1  at  the  point  (1,  1). 


264 


INFINITESIMAL    CALCULUS. 


[Ch.  XVIL 


144.   Osculating  circle.     It  was  pointed  out  in  Art.  143,  Note  3, 

that  contact  of  the  second  order  is,  in  general,  the  closest  contact 

that  a  circle  can  have  with  a 
curve.  A  circle  having  contact 
of  the  second  order  with  a  curve 
at  a  point  is  called  the  osculating 
circle  at  that  point. 

In  Fig.  83  PT  is  tangent  to  the 
curve  C  at  P.  Every  circle  which 
passes  through  P  and  has  its  cen- 
tre in  the  normal  AOTf  touches  C 
at  P.  One  of  these  circles  has 
contact  of  the  second  order  with 
(7  at  P;  let  this  circle  be  denoted 

by  K.     All  the  other  circles,  infinite  in  number,  in  general  have 

contact  of  the  first  order  only. 

Osculating  circle:  rectangular  coordinates.     The  radius  and  the 

centre  of  the  osculating  circle  at  any  point  P{x,  y)  on  the  curve 


Fig.  83. 


y=f(^) 


(1) 


will  now  be  obtained.     Denote  the  centre  and  radius  by  (a,  b) 
and  r.     Then  the  equation  of  the  osculating  circle  at  the  point 

^^'^)'^  (X-ay+(Y-by  =  i^.  (2) 


For  the  moment,  for  the  sake  of  distinction,  x  and  y  are  used 

to  denote  the  coordinates  of  a  point  on  tlie  curve,  and  X  and  Y 

are  used  to  denote  the  coordinates  of  a  point  on  the  circle.     Then 

at  the  point  where  the  circle  and  the  curve  have  contact  of  the 

second  order  ,  _^       .       ., ,_       ., 

dY  _  dy    d-Y  _  fn/^ 

dX^dx    dX^~d?' 


X 


y, 


(3) 


From  (2),  on  differentiating  twice  in  succession, 
X 


„H.(r-6)||=o, 


\dXj      ^  'dX' 


(4) 
(5) 


144,  145.] 


CURVATURE. 


265 


and 


.-.  F-6=- 


X-a  = 


dYyndY  ^  cV 
dx)  \dX  '  d: 


Y 

'  dX^' 


Accordingly,  from  (3),  (2),  (6),  (7), 


[^ -(!)?. 


and  from  (3),  (6),  (7), 


\dx]       dy     , 
d-iy  dx '  ^ 

d3^ 


dxi 


(6) 
(7) 

(8) 
(9) 


Note.     For  the  osculating  circle,  polar  coordinates  being  used,  see  Art. 
150,  Note  2. 

Ex.  1.    Determine  the  radius  and  the  centre  of  the  osculating  circle  for 
each  of  the  curves  in  Ex.  1  (1),  Art.  143,  at  their  point  of  contact. 

Ex.  2.    Do  as  in  Ex.  1  for  the  curves  Ex.  1  (2),  Art.  143. 

145.  The  notion  of  curvature.  Let  the  curves  A,  B,  C,  D  have 
a  common  tangent  FT  at  P.  At  the  point  P  the  curve  A,  to  use 
the  popular  phrase,  bends  or  curves  more  than  the  curves  B,  C, 
and  D ;  and  D  bends  or  curves  less  than  the  curves  A,  B,  and  C. 
These  four  curves  evidently  differ  in  the  rate  at 
which  they  bend,  or  turn  away  from  the  straight 
line  PT,  at  P.  These  ideas  are  sometimes  ex- 
pressed by  saying  that  these  curves  differ  in 
curvature  at  P,  and  that  there  A  has  the  greatest 
and  D  the  least  curvature.  In  the  case  of  two 
circles,  say  one  with  a  radius  of  an  inch  and  the 
other  Avith  a  radius  of  a  million  miles,  it  is  cus- 
tomary to  say  that  the  secoad  circle  has  a  small 
curvature,  and  that  the  firsfehas  a  large  curvature  in  comparison 
with  the  second.  An  inspection  of  a  figure  consisting  of  a  circle 
and  some  of  its  tangents  gives  the  impression  that  what  is  popu- 
larly called  the  curvature  is  the  same  at  all  points  of  that  circle. 


Fig.  84. 


266 


INFINITESIMAL   CALCULUS. 


[Ch.  XVIL 


On  the  other  hand,  an  inspection  of  an  elongated  ellipse  gives 
the  impression  that  the  curvature  is  not  the  same  at  all  points 
of  that  ellipse,  although  at  two  particular  points,  or  at  four 
particular  points,  it  may  be  the  same.  Curvature  will  now  be 
given  a  precise  mathematical  definition  and  its  measurement 
will  be  explained. 

Ex.  1.  Draw  an  ellipse,  and  find  by  inspection  the  points  where  the  curva- 
ture is  greatest  and  where  it  is  least.  Show  how  to  obtain  sets  of  four  points 
on  the  ellipse  which  have  the  same  curvature. 

Ex.  2.    Discuss  a  parabola  and  an  hyperbola  in  the  manner  of  Ex.  1. 

146.   Total  curvature.     Average  curvature.     Curvature  at  a  point. 

At  Ai  the  curve  C  has  the  direction  A^T^,  which  makes  the  angle 

<^i  with  the  a;-axis ;  at  A2  the 
curve  has  the  direction  A2T.2, 
which  makes  an  angle  ^2  with  the 
aj-axis.  The  difference  between 
these  directions  represents  the 
angle  by  which  the  curve  has 
changed  its  direction  from  the 
direction  of  the  line  A^T^  in 
the  interval  of  arc  from  A^  to 
Ao.  This  difference,  namely, 
T1RT2  or  cf)2  —  <f>i,  is  called  the 
total  curvature   of  the  arc  A1A2. 

The  average  curvature  for  this  arc  is 

(<^2  —  <^i)  ^  length  of  arc  ^1^2- 
(Here  the  angle  is  measured  in  radians.) 

Accordingly,  if  (Fig.  86)  A<^  is  the  angle  between  the  tangents 
at  A  and  B,  then  A<^  is  the  total  curva-         y 
ture  of  the  arc  AB-,  if  As  is  the  length 

of  the  arc  AB,  then  — ^  is  the  average 

As 

curvature  of  that  arc.  Now  let  B 
approach  A.  The  arc  As  and  the  angle 
A<^    then    become    infinitesimal ;    and, 

finally,  when  B  reaches  A,  — ^  has  the    ^0 

As  Fig.  86. 


146-148.]  CURVATURE.  267 

limiting  value  -^.     The  limit. ,^o—  at  any  point  on  a  curve,  i.e. 
ds  As 

^  there,  is  called  the  curvature  of  the  curve  at  that  2»oint.     (The 

phrase  "  curvature  of  a  curve  "  means  the  curvature  of  the  curve 
at  a  particular  point.)  In  all  curves,  with  the  exception  of 
straight  lines  and  circles,  the  curvature,  in  general,  varies  from 
point  to  point. 

147.  The  curvature  of  a  circle.  Let  A  and  B  be  two  points  on 
a  circle  having  its  centre  at  0.  In 
Fig.  87  the  angle  between  the  direc- 
tions of  the  tangents  AT^  and  BT^  is 
A<^,  say.  Let  As  denote  the  length  of 
the  arc  ^5.  ThQXi  AOB=T^ET,=:^<^. 
Hence,  by  trigonometry.  As  =  rA<^. 

From  this, 

^^1.  whence    ^-l.        (\\ 

^s       r'  ^^^"^"^^    ds~r  ^  ^  Fig.  87. 

That  is,  the  curvature  of  a  circle  is  constant  and  is  the  reciprocal 
of  {the  measure  of)  the  radius. 

Note.  When  the  radius  increases  beyond  all  bounds,  the  curvature 
approaches  zero,  and  the  circle  approaches  a  straight  line  as  its  limiting 
position.  When  the  radius  decreases,  the  curvature  increases ;  as  the  radius 
approaches  zero  and  the  circle  thus  shrinks  towards  a  point,  the  curvature 
approaches  an  infinitely  great  value. 

It  is  shown  in  Ex.  6,  Art.  194,  that  all  curves  of  constant  curvature  are 
circles. 

Ex.  Compare  the  curvatures  of  circles  of  radii  2  inches,  2  feet,  5  yards, 
2  miles,  10  miles,  100  miles,  and  1,000,000  miles. 

148.  To  find  the  curvature  at  any  point  of  a  curve :  rectangular 
coordinates.  Let  the  curve  in  Fig.  ^Q>  be  y=f(x),  and  let  its 
curvature  at  any  point  A(x,  y)  be  required.  Let  k  denote  the 
curvature  at  A,  and  <^  denote  the  angle  which  the  tangent  at  A 
makes  with  the  a;-axis.  Take  an  arc  AB  and  denote  its  length 
by  As,  and  denote  the  angle  between  the  tangents  at  A  and  B  by 
A<^.     Then,  by  the  definition  in  Art.  146, 

k  =  ^^tA. 
as 


268  INFINITESIMAL   CALCULUS.  [Ch.  XVII. 

Now  (Art.  58),         tan  6  =  ^-     .-.  <f>  =  tan"^  ^. 
^  ^      dx  ^  dx 

d^ 


ds      dsV  dx/      da;V  d.17     ffo"      i   ,   /*y V  '   dx 


[Art.  67  c(2)],  fc  = ^ 


(1) 


i^-am 


This,  by  (1)  Art.  147  and  (8)  Art.  144,  is  the  same  as  the  curva- 
ture of  the  osculating  circle. 

In  order  to  find  the  curvature  at  a  definite  point  (xi,  y{)  it  is 
only  necessary  to  substitute  the  coordinates  Xi,  y^,  in  the  general 
result  (1), 

Ex.  1.  Compute  and  compare  the  curvatures  of  the  two  curves  in  Ex.  1  (1), 
Art.  143,  at  their  point  of  contact. 

Ex.  2.  Find  the  curvature  of  the  curve  ?/  =  ic*^  —  2x''  +  7  ic  at  the  origin. 
Determine  the  radius  and  centre  of  its  osculating  circle  at  that  point. 

149.  The  circle  of  curvature  at  any  point  on  a  curve  :  rectangular 
coordinates.  The  circle  of  curvature  at  a  point  on  a  curve  is  the 
circle  which  passes  through  the  point  and  has  the  same  tangent 
and  the  same  curvature  as  the  curve  has  there.  The  radius  of 
this  circle  is  called  the  radius  of  curvature  at  the  point,  and  the 
centre  of  the  circle  is  called  the  centre  of  curvature  for  the  point. 

The  radius  of  curvature.  Let  It  denote  the  radius  of  curvature 
and  (a,  p)  denote  the  centre  of  curvature  for  any  point  (x,  y)  on 
the  curve  y  =/(x).  Then  it  follows  from  Art.  147,  and  Art.  148, 
Eq.  1,  that  ^ 

(That  is,  R  is  the  value  of  this  expression  at  that  point.) 

Note  1.  There  is  an  infinite  number  of  circles  that  can  pass  through  a 
given  point  on  a  curve  and  have  the  same  tangent  as  the  curve  has  there  but 
not  the  same  curvature,  and  there  is  an  infinite  number  of  circles  that  can 


149.] 


CIRCLE  OF  CVkVATVRE. 


269 


pass  through  this  point  and  have  the  same  curvature  but  not  the  same  tangent 
as  the  curve  has  there ;  but  tliere  is  only  one  circle  passing  through  the  point 
that  has  there  both  the  same  tangent  and  the  same  curvature  as  the  curve. 

Ex.  1.    Illustrate  Note  1  by  figures. 

The  centre  of  curvature.     Since  at  any  point  on  a  curve  the  circle 
of  curvature  and  the  curve  have  the  same  tangent  and  curvature, 

it  follows  that  —  and  ^.  are  respectively  the  same  for  the  circle 

and  the  curve  at  that  point.  Accordingly  (Art.  143,  Note  3)  the 
circle  of  curvature  has,  in  general,*  contact  of  the  second  order 
with  the  curve,  and  thus  (Art.  144)  coincides  with  the  osculating 
circle  passing  through  the  point.     Accordingly  (Art.  144,  Eq.  9) 


dJl. 
doc' 


P  =  2/  + 


d^y 
dx^ 


(2) 


Note  2.     The  coordinates  of  the  centre  of  curvature  may  also  be  obtained 
in  the  following  manner. 

Let  C  be  the  centre  of  the  circle  of  cur- 
vature of  the  curve  TL  at  P,  and  let  the 
tangent  FT  make  the  angle  ^  with  the 
X-axis.  Draw  the  ordinates  TM  and  CN, 
and  draw  PB  parallel  to  OX.  Let  J? 
denote    the    radius    of    curvature.      Then 


NCF  =  0,  and  tan  0  = 


_dy^ 


dx 


In  Fig.  88 
a=  O.V  = 


OM-  BP=x-  Rs'm(f> 


b-m' 


cly 
dx 


dJC2 


[-(l)t 


N/T      M 
Fig.  88. 

d^y  dx 


dx^ 


14- 


Also,    /3  =  NC  =  MP  +  BC  =  y-\-  B  cos  <p  =  y  -\ 
The  results  for  Fig.  88  are  true  for  all  figures. 


(dyV 
\dxl 


d^ 
dx'^ 


(3) 


(4) 


*  For  an  exception  see  the  circles  of  curvature  at  the  ends  of  the  axes  of 
an  ellipse.     (See  Ex.  4  following. ) 


270  INFINITESIMAL   CALCULUS.  [Ch.  XVII. 

Ex.  2.  Verify  the  last  statement  by  drawing  the  radii  of  curvature  at  points 
on  each  side  of  points  of  maximum  and  minimum  in  the  curves  in  Fig.  80 

and  carefully  noting  the  algebraic  signs  of  ^  and  — ^  at  these  points. 

dx  d^x 

Note  3,  A  glance  at  Fig.  80  shows  that  at  P  and  li  the  normal  (Art.  59) 
and  the  radius  of  curvature  have  the  same  direction,  and  at  Q  and  S  they 
have  opposite  directions.  Hence  (see  Art.  142)  the  normal  and  the  radius  of 
curvature  at  a  point  on  a  curve  have  the  same  or  opposite  directions  accord- 

d^v 
ing  as  y — ^  there  is  respectively  negative  or  positive. 
dx^ 

Note  4.    At  a  point  of  inflexion,  according  to  Art.  78,  and  Art.  148,  Eq.  (1), 

the  curvature  is  zero. 

Note  5.  A  centre  of  curvature  is  the  limiting  position  of  the  intersection 
of  two  infinitely  near  normals  to  the  curve.  For  a  consideration  of  this  im- 
portant geometrical  fact,  see  Williamson,  Diff.  Cal.  (7th  ed.),  Art.  229; 
Lamb,  Calculus.,  Art.  150  ;  Gibson,  Calculus.,  Art.  141. 


EXAMPLES. 

3.  Find  the  radius  of  curvature  and  the  centre  of  curvature  at  any  point 
on  the  parabola  y'^  =  ipx.     What  are  they  for  the  vertex  ? 

Apply  the  general  results  just  obtained  to  particular  cases,  by  giving  p  par- 
ticular values,  e.g.  1,  2,  etc.,  and  taking  particular  points  on  the  curves, 
and  make  the  corresponding  figures. 

N.B.  As  in  Ex.  8,  apply  the  general  results  obtained  in  the  following 
examples  to  particular  cases. 

4.  As  in  Ex.  3  for  the  ellipse  b^x^  +  a^y^  =  a^b^.  Find  the  radii  of  cur- 
vature at  the  ends  of  the  axes.  Show  that  this  radius  at  an  extremity  of 
the  major  axis  is  equal  to  half  the  latus  rectum.  Illustrate  Note  4,  Art.  143, 
by  drawing  an  ellipse  and  the  circles  of  curvature  at  various  points  on  it. 
Show  that  the  circles  of  curvature  for  an  ellipse,  at  the  ends  of  the  axes,  have 
contact  of  the  third  order  with  the  ellipse. 

5.  Find  the  radius  and  centre  of  curvature  at  any  point  of  each  of  the  fol- 
lowing curves :    (1)  The  hyperbola  b'^x^  —  a~y'^  =  a'^b^.     (2)  The  hyperbola 

xy  =  a2.     (3)  The  catenary  y  =  ^  {e^  +  e~«).     (4)  The  astroid  x^  ^  ij^  =  ai 

(5)  The  astroid  x  =  aco^^d,  y  =  as\n^d.  (6)  The  semi-cubical  parabola 
x^  =  ay'^.  (7)  The  curve  x^y  =  a^{x  -  y)  where  x  =  a.  (8)  The  cycloid 
X  =  a{6  —  sin ^),  y  =  a(l  —  cos 6).  In  this  cycloid  show  that  the  length  of 
the  radius  of  curvature  at  any  point  is  twice  the  length  of  the  normal. 

6.  Find  the  radius  of  curvature  at  any  point  of  each  of  the  following 
curves  :  (1)  The  parabola  Vx  +  Vy  =  Va.  In  this  curve  show  that  a  +  ^3  = 
S(x  +  y) .     (2)  The  cubical  parabola  a'^y  =  x\     (3)  The  catenary  of  uniform 


150.]  CIRCLE  OF  CURVATURE.  271 

strength  ij  =  clog  sec  [-).      (4)   The  witch  xij- =  a'(a  -  x)  at  the  vertex. 

(5)  The  parabola  x  =  a  coV^xp,  y  =  2  acotrj/.  (0)  The  ellipse  x  =  a  cos0, 
y  =  b  sin  0.  (7)  The  hyperbola  x  =  asec((>,  y  =  b  tan  <p.  (8)  The  catenary 
x  =  a  log  (sec  d  +  tan  6),  y  =  a  sec  ^. 

150.  The  radius  of  curvature  :  polar  coordinates.  This  can  be  deduced 
(a)  directly  from  the  definition  of  curvature  (Art.  146)  and  the  definition  of 
racWus  of  curvature  (Art.  149);  and  (&)  from  form  (1),  Art.  149,  by  the 
usual  substitution  for  transformation  of  coordinates,  namely,  x  =  r  cos  0, 
y  =  r  sin  6. 

(a)    By  Art.  60  (2),  0  =  ^  +  ,/,. 

Now  k  =  '1^  (Art.  146)  ='^^  . 'Il  =  (^ +m  [r^  +  (^YY^. 

ds  ^       dd     ds      \        d^y  L         \dd)  J 

[Art.  67  d,  Eq.  (3).] 
(dry     .  d^r 

_  ^     .  _^WW 

dr  \       "  dd 


fl^  ..   .   at^^        .    /  _  .      -i  (  r  ]         ^  d^       \dd)        '  dff^ 


Also,  tan  xp  =  r  ^^  (Art.  60).     .-.  \p  =  tan-i 


x_  P-(i)t 


(7:?  J 


r-2  + 


(!) 


Hence      B  =  j  =  —^ — n  r^.- iT'  (1) 


(6)   The  deduction  of  (1)  from  (1),  Art.  149,  by  the  transformation  of  coor- 
dinates is  left  as  an  exercise  for  the  student.  /  ,  x  o   4 

["^+(I)J 


Note  1.     On  the  substitution  of  u  for  -  in  (1),  J?  =  -; ^ — - — 


dd-^j 
Note  2.     Since  the  osculating  circle  and  the  circle  of  curvature  coincide, 

the  forms  just  found  for  B  give  the  radius  of  the  osculating  circle. 

Note  .3.     For  other  expressions  for  R  see  Todhunter,  Diff.  Cat,  Art.  321, 

and  Ex.  4,  page  352  ;  Williamson,  Diff.  Cal.  (7th  ed.),  Art.  236.     Also  see 

F.  G.  Taylor,  Calculus,  Arts.  288-290. 

EXAMPLES.  * 

1.  Find  the  radius  of  curvature  at  any  point  of  each  of  the  following 
curves  :  (1)  The  circles  r  =  a  and  r  =  2  &  cos  ^.  (2)  The  parabola  r(l  +  cos  d) 
=  2  a.  (3)  The  cardioid  r  =  a(l  +  cos  d).  (4)  The  equilateral  hyperbola 
?'2 cos 2 (?  =  «"^.  (5)  The  lemniscate  r'^  =  a^cos2  0.  (6)  The  logarithmic 
spiral  r  =  e«».  (7)  The  spiral  of  Archimedes  r  =  ai>.  (8)  The  general 
spiral  r  —  a<p^. 

2.  Derive  the  expression  for  R  in  Note  1. 


272 


INFINITESIMAL   CALCULUS. 


[Ch.  XVII. 


151.  Evolute  of  a  curve.  Corresponding  to  each  point  on  a 
given  curve  there  is  a  centre  of  curvature.  The  locus  of  the 
centres   of   curvature  for  all  the  points  on  the  curve,  is  called 

the  evolute  of  the  curve. 
Thus,  if  AA^  be  the 
given  curve  and  Cj, 
^2)  C3,  •••,  be  respec- 
tively the  centres  of 
curvature  for  any 
points  A^,  A.2,  A^,  •••, 
on  the  given  curve, 
the  curve  CiCoCs  is 
the  evolute  of  AA^. 

To  find  the  equation 
of  the  evolute  of  the 
curve.  Let  the  equa- 
tion of  the  given 
curve  be 

y=f{x),      (1) 

and  let  A{x,  y)  be  any  point  on  it.     Let  C  be  the  centre  of  curva- 
ture for  the  point  A,  and  denote  C  by  («,  ^).     Then  [Art.  149, 

^dx)     dy 


X  —  a 


y-P  =  - 


d^y 

d'y 
dx' 


dx 


(2) 


(3) 


On  the  elimination  of  x  and  y  from  equations  (1),  (2),  (3),  there 
will  appear  an  equation  which  is  satisfied  by  a  and  /8,  the  coordi- 
nates of  the  point  C.  But  A  is  any  point  on  the  given  curve,  and, 
accordingly,  C  is  any  of  the  centres  of  curvature  for  the  points  on 
AAi.  Accordingly,  the  equation  found  as  indicated  is  the  equa- 
tion of  the  evolute. 

Note.  The  algebraic  process  of  eliminating  x  and  y  from  (1),  (2),  and 
(3)  depends  on  the  form  of  these  equations. 


15 1,  152.]  THE  EVOLUTE.  273 

EXAMPLES. 

1.  Find  the  evolute  of  the  parabola 

y''  =  4.px.  (1) 

Here  by  Ex.  3,  Art.  149,  a  =  2  p  +  3  x  ;  (2) 

The  elimination  of  x  and  y  between  equations  (1),  (2),  (3),  gives  the 
equation  of  the  evolute,  viz.  the  semi-cubical  parabola 

4(a-2i?)3  =  27pi32; 

i.e.  on  using  the  ordinary  notation  for  the  coordinates, 

4(X-2j9)3zz:27pi/2, 

2.  Find  the  evolute  of  the  ellipse  fe%2  +  ^V^  =  a'^h'^.  (1) 
Here,  by  Ex.  4,  Art.  149,                a  =  {^^^-=^\x\  (2) 

_,  =  («^)^.  (3) 

The  elimination  of  x  and  y  between  equations  (1),  (2),  (3),  gives  the  equa.- 
tion  of  the  evolute,  viz.  : 

{aa)i  +  (6/3)1  =  (a2  -  62) |^ 
i.e.  on  using  the  ordinary  notation  for  coordinates, 

iax)i  +  (6j/)f  =  (a2  -  62)1. 

3.  Find  the  evolute  of  the  following  curves  :  (1)  the  hyperbola  62^2  _  g^y^ 
=  a262.  (2)  The  equilateral  hyperbola  xy  =  a^.  (3)  The  four-cusped  hypo- 
cycloid  a;3  -f  ^¥  =  as. 

4.  Find  both  geometrically  and  analytically  the  evolute  of  a  circle. 

5.  Show  that  the  evolute  of  a  complete  arch  of  a  cycloid  consists  of  the 
halves  of  an  equal  cycloid.     [Suggestion  :  see  Ex.  5  (8),  Art.  149.] 

152.  Properties  of  the  evolute.  The  two  most  important  proper- 
ties of  the  evolute  of  a  curve  are  the  following : 

(a)  Tlie  normal  at  any  j^oint  of  a  given  cui-ve  is  a  tangent  to  the 
evolute,  and  any  tangent  to  the  evolute  is  a  normal  to  the  given  curve. 

(6)  The  length  of  an  arc  of  an  evolute,  provided  that  the  curva- 
ture varies  continuously  from  point  to  point  along  this  arc,  is 
equal  to  the  difference  between  the  lengths  of  the  two  radii  of  curvature 
draimi  from  the  given  curve  to  the  extremities  of  the  arc. 


274 


IN  FIN  I TESIMA  L   CA  LCULUS. 


[Ch.  XVII. 


Proof  of  (a).  Let  AA^  (Fig.  89)  be  the  given  curve,  and  let  its 
equation  be  y  =  f{x),  and  let  CC^  be  its  evolute.  Let  (7(«,  /S)  be 
the  centre  of  curvature  for  any  point  A{x,  y). 

The  slope  of  the  given  curve  at  A  is  -^,  and  the  slope  of  the 
evolute  at  C  is    ^-     From  Equations  (2),  Art.  149,  on  differentia- 


tion and  reduction, 


^dyfd^y 
d/3_    dx\dafj 
dx 


\dxj  Jc?aJ^ 


da 

dx 


fd^V 
[clx^J 

_dyf^dyrd^\^ 
dx  [    dx\dx- 


■HS 


dx^} 


dx^ 


From  (1)  and  (2),  and  Art.  34  (3) 
d^_fd^_^da 


d^     \dx     dx 


doc 
dy 


(1) 


(2) 


(3) 


dx 


But  —  ^  is  the  slope  of  the  normal  at  A(x,  y).  '  Hence,  the 
normal  at  A  and  the  tangent  to  the  evolute  at  C  coincide. 

Y 


fA   X 


Fig.  90  a. 


Fig.  90  6. 


Note  1.  Thus,  in  Fig.  89,  AC  i^  the  radius  of  curvature  for  A  on  J.^i, 
AC  is  normal  to  AAi  at  A^  and  AC  touches  the  evolute  CCi  at  C.  In  Figs. 
90  a,  90  &,  PiCi,  P-zCo,  are  normal  to  the  parabola  and  tangent  to  its  evolute ; 
PC  is  normal  to  the  ellipse  and  tangent  to  its  evolute. 


152.] 


THE  E VOLUTE. 


275 


Note  2.  On  account  of  property  (a)  the  evolute  is  sometimes  defined  as 
the  envelope  (see  Art.  154)  of  the  normals  of  the  curve.  See  Art.  157  (Ex. 
2  and  Notes  4,  5)  and  Art.  158,  Ex.  1.  Also  see  Echols,  Calculus,  Arts. 
106-108. 

Proof  of  (&).  Let  K  be  the  given  curve  y=f(x),  and  E  its 
evolute. 

Let  Oi  be  the  centre  of  curva- 
ture for  Ai,  and  C2  the  centre  of 
curvature  for  Ao.  Denote  any 
point  in  K  by  (x,  y),  the  radius 
of  curvature  there  by  E,  and  the 
corresponding  centre  of  curvature 
in  E  by  (a,  /8).  Let  the  points  Ai, 
A2y  C],  C2,  be  denoted  as  {x-^,  y^, 
(ajg,  2^2),  («i,  A),  («2,  /?2),  respec- 
tively; also  let  the  radii  of  cur- 
vature A^C^  and  A^C^  be  denoted 
by  jRi  and  R2.     It  will  now  be  shown  that 

length  of  arc  C1C2  =  jB2  -  ^1 


^2.^2-!/2) 


Fig.  91. 


Arc  CiC2=   P  ^'-Jl  +  f— Y-f7)8.     (See  Art.  137.)      (4) 

On  substituting  the  value  of  —  from  (3),  and  the  value  of  d^ 
derived  from  (1),  and  noting  that 

X  =  a'l  when  /3  =  fti,  and  x  =  X2  when  ^  =  ^^^ 


Equation  (4)  becomes 


arc  CiCg  = 


rw 


1  + 


dxydx")      \_       \clxj  Jdx^ 


\dx.  (5) 


Differentiation  of  R  in  Art.  149,  Eq.  (1),  will  show  that  —  is 

fix 
the  same  as  the  integrand  in   (5).      Then,   since  R=  R^  when 

dR 

x  =  £Ci,  and  R  =  R.,  when  x  =  X2,  and  —  dx  =  dR  (Art.  27),  Equa- 

(XX 


tion  (5)  becomes 

'^dx=l        dR=l         dR:^R2-R, 

=xi       dx  a/x=xi  %J  R=Ri 


arc 


276  INFINITESIMAL    CALCULUS.  [Ch.  XVII. 


Note  3.     See  Echols,  Calculus,  Art.  170  and  Chap.  XIV. 
Ex.  1.    Show  that  the  total  length  of  the  evolute  of  the  ellipse  whose 
semi-axes  are  a  and  h,  is 


4(^3 


ah 

Ex.  2.  Show  that  the  length  of  the  evolute  of  the  parabola  ?/2  =  4|)x  that 
is  intercepted  by  the  parabola  {i.e.  2  SB,  Fig.  90  a)  is  4p  (3\/3  —  1). 

153.  Involutes  of  a  curve.  In  Fig.  89  the  curve  CC^  is  the 
evolute  of  the  curve  AA^.  Suppose  that  a  string  is  stretched 
tightly  along  the  curve  CCj  and  held  taut  in  the  position 
LC1C2C.3C,  the  portion  LCi  thus  being  tangent  to  the  evolute 
at  Ci.  Now,  a  point  A^  being  taken  in  the  string,  let  it  be 
unwound  from  OjC  It  follows  from  properties  (a)  and  (6), 
Art.  152,  that,  as  the  string  is  unwound  from  the  evolute  C^C,  A^ 
will  describe  the  curve  A^A.  It  is  on  account  of  this  property 
that  CCi  is  called  the  evolute  of  AA^.  On  the  other  hand,  AA^ 
is  called  an  involute  of  CCi.  "An  involute,"  because  CCi  has  an 
infinitely  great  number  of  involutes.  For,  when  the  string  is 
unwound  from  the  evolute  CiC  an  involute  will  be  traced  out 
by  each  point  like  Ai  taken  in  the  string  LA-^CiCc^C^.  These 
involutes  are  parallel  curves  *  ;  for  (1)  they  have  the  same  normals, 
namely,  the  tangents  of  their  common  evolute,  and  (2)  the  dis- 
tance between  any  two  of  them  along  these  normals  is  constant, 
namely,  the  distance  between  the  two  points  originally  taken  on 
the  string  that  is  being  unwound.  Figure  89  shows  three  involutes 
of  CCi. 

EXAMPLES. 

1.  Construct  several  involutes  of  the  evolute  of  the  parabola  whose  latus 
rectum  is  8  (besides  the  parabola  itself). 

2.  Construct  several  involutes  of  the  evolute  of  the  ellipse  whose  axes 
are  9  and  25. 

3.  Given  a  cycloid,  construct  the  involute  that  is  traced  out  by  the  point 
at  the  vertex  in  the  course  of  "  the  unwinding." 

4.  Given  a  circle,  construct  the  involute  that  is  traced  out  by  any  point 
on  the  circle  in  the  course  of  "the  unwinding."  (In  the  case  of  a  circle 
all  such  involutes  are  identically  equal.  Accordingly,  such  an  involute  is 
usually  termed  "  the  involute  of  the  circle.") 

5.  Construct  several  involutes  of  an  ellipse,  and  several  involutes  of  a 
parabola. 

*  Two  curves  are  said  to  be  parallel  when  they  have  common  normals 
always  differing  in  length  by  the  same  amount. 


CHAPTER   XVIII. 


SPECIAL   TOPICS   RELATING  TO    CURVES. 
ENVELOPES,  ASYMPTOTES,  SINGULAK  POINTS,  CURVE  TRACING. 

Envelopes. 

154.   Family  of  curves.     Envelope  of  a  family  of  curves.     The 

idea  of  a  family  of  curves  may  be  introduced  by  an  example. 
The  equation  /         x2  ,     2      ^ 


(x-cf-{-y'-  =  4. 


is  the  equation  of  a  circle  of  radius  2  whose  centre  is  at  (c,  0). 
If  c  be  given  particular  values,  say  2,  3,  —0,  the  equations  of 
particular  circles  are  obtained.  Thus  Equation  (I)  really  repre- 
sents a  family  of  circles,  viz.  the  circles  (see  Fig.  92)  whose  radii 


Fig.  92. 

are  2  and  whose  centres  are  on  the  a>axis.  The  individual 
members  of  the  family  are  obtained  by  letting  c  change  its  values 
from  —  00  to  -f  00.  A  number  such  as  c,  whose  different  values 
serve  to  distinguish  the  individual  members  of  a  family  of  curves, 
is  called  the  parameter  of  the  family.  Thus,  to  take  another 
example,  the  equation  y  =  2x-\-b  represents  the  family  of  straight 
lines  having  the  slope  2 ;  and  y  =  2  x-\-o,  y  =  2x  —  l,  are  particu- 
lar lines  of  the  family.  (Let  a  figure  be  constructed.)  In  this 
case  the  parameter  h  can  take  all  values  from  —  00  to  +  oo. 

277 


278  INFINITESIMAL    CALCULUS.  [Ch.  XVIIl. 

To  generalize :  f(x,  y,  a)  =  0  (2) 

is  the  equation  of  a  family  of  curves  whose  parameter  is  a.  The 
individual  members  or  curves  of  the  family  are  obtained  by  giving 
particular  values  to  a.  These  curves  are  all  of  the  same  kind, 
but  differ  in  various  ways ;  for  instance,  in  position,  shape,  or 
enclosed  area.  A  family  of  curves  may  have  two  or  more  param- 
eters. Thus,  y  =  mx  -\-  b,  in  which  m  and  b  may  take  any  values, 
has  two  parameters  m  and  b,  and  represents  all  lines.  The  equa- 
tion {x  —  hy  -\-  (y  —  ky  =  25,  in  which  h  and  k  may  take  any 
values,  represents  all  circles  of  radius  5.  The  equation  (x  —  lif 
-\-{y  —  lif  =  r"^,  in  which  li,  k,  and  r  may  each  take  any  value, 
represents  all  circles. 

Envelope.  The  envelope  of  a  family  of  curves  is  the  curve,  or 
consists  of  the  set  of  curves,  which  touches  every  member  of  the 
family  and  which,  at  each  point,  is  touched  by  some  member  of 
the  family.  For  example,  the  envelope  of  the  family  of  circles 
in  Fig.  92  evidently  consists  of  the  two  lines  y  —  2=0  and  y-^2  =  0. 
On  the  other  hand,  the  family  of  parallel  straight  lines  y=2x-\-b 
does  not  have  an  envelope  ;  and,  obviously,  a  family  of  concentric 
circles  cannot  have  an  envelope. 

EXAMPLES. 

1.  Say  what  family  of  curves  is  represented  by  each  of  the  following 
equations,  and  in  each  instance  make  a  sketch  showing  several  members  of 
the  family : 

(a)  x^  +  ?/2  =  r^,  parameter  r.         (6)    y  =  mx  +  4,  parameter  m. 
(c)  y^  =  4|)x,  parameter  p.  (d)   y^  =  i  a(x  4-  a),  parameter  a. 

(e)   — I-  2_  =  1,  parameter  a.  (  f)  — '■ 1 -'- —  =  1,  parameter  k. 


(g)  y  =  mx  H — ,  parameter  m.       (h)    y  =  mx  +  V25  m'  +  10,  parameter  m. 
m 

2.    Express  opinions  as  to  which  of  the  families  in  Ex.  1  have  envelopes, 

and  as  to  what  these  envelopes  may  be. 

155.  Locus  of  the  ultimate  intersections  of  the  curves  of  a  family. 
In  Eq.  (2),  Art.  154,  the  equation  of  a  family  of  curves,  let  a  be 
given  the  particular  value  a^ ;  then  there  is  obtained  the  equation 
of  a  particular  member  of  that  family,  viz. 

f{x,y,a,)  =  0.  (1) 


155.]  ENVELOPES.  279 

Also,  f{x,  y,  a-i  -i-h)  =  0 

is  the  equation  of  another  member  of  the  family.  Let  I.  and  II. 
be  these  curves.  The  smaller  h  becomes,  the  more  nearly  does 
curve  II.  come  into  coincidence  with  curve  I.  Moreover,  as  h  be- 
comes smaller  and  approaches  zero,  A,  the  point  of  intersection  of 
these    curves,    approaches    a 

definit-e  limiting  position.    For  ^u.V;^ 

example,  if  (Fig.  92)  the  centre 
L  approaches  nearer  to  C,  then 
K,  the  point  of  intersection  of 
the  circles  whose  centres  are 

at  C  and  L,  moves  nearer  to       j^  Fig.  93. 

P;  and  finally,  when  L  reaches 

C,  ^arrives  at  the  definite  position  P.  The  locus  of  the  limiting 
position  of  the  point  (or  points)  of  intersection  of  two  curves  of 
a  family  which  are  approaching  coincidence  is  called  the  locus  of 
ultimate  intersections  of  the  curves  of  the  family.  For  instance,  in 
the  case  of  the  family  of  circles  in  Fig.  92,  this  locus  evidently 
consists  of  the  lines  y  —  2  =  0  and  y  -\-2  =  0. 

Note.     The  last-mentioned  locus  may  also  be  derived  analytically. 

Let  (a;  _  ci)2  +  ?/2  =  4  (1) 

and  (x-cx-hy-vy'^  =  ^  (2) 

be  two  of  the  circles.  On  solving  these  equations  simultaneously  in  order  to 
find  the  point  of  intersection,  there  is  obtained 

(x  -  ci)2  _  (x  -  ci  -  hY  =  0  ;   whence  A(2  a:  -  2  Ci  -  /i)  =  0, 
and,  accordingly,  x  —  Cx-\ — 

An  ultimate  point  of  intersection  is  obtained  by  letting  h  approach  zero. 
If  ^  =  0,  then  X  =  C\,  and  by  (1)  ?/  =  ±  2.  Thus  y  =z±2  at  the  ultimate 
points  of  intersection,  and  therefore  the  locus  of  these  points  is  the  pair  of 
lines  y  =  ±2. 

N.B.  In  the  following  articles  "the  locus  of  ultimate  intersections"  is 
denoted  by  I.  u.  i. 


280  INFINITESIMAL   CALCULUS.  [Ch.  XVIII. 

156.  Theorem.  In  general,  the  locus  of  the  ultimate  intersections 
touches  each  member  of  the  family.  Let  I.,  II.,  III.  be  any  three 
members  of  the  family,  and  let  I.  and  II.  intersect  at  F,  and  11. 
and  III.  at  Q.  When  the  curve  I.  approaches  coincidence  with 
II.,  the  point  F  approaches  a  definite  position  on  l.  u.  i.  of  the 
curves  of  the  family.  When  the  curve  III.  approaches  coincidence 
with  II.,  Q  approaches  a  definite  position  on  I.  u.  i.  When  I.  and 
III.  both  approach  coincidence  with  II.,  F  and  Q  approach  each 
other  along  II.,  and  at  the  same  time  approach  I.  u.  i.     When  F 


and  Q  finally  reach  each  other  on  II.,  they  are  also  on  I.  u.  i.  More- 
over, when  F  and  Q  come  together,  the  tangent  to  II.  at  F  and  the 
tangent  to  II.  at  Q  come  into  coincidence  as  a  line  which  is  at  the 
same  time  a  tangent  to  curve  II.  and  a  tangent  to  t  u.  i.  at  the  point 
where  F  and  Q  meet.  Thus  the  curve  II.  and  I.  u.  i.  have  a  com- 
mon tangent  at  their  common  point.  Similarly  it  can  be  shown 
that  I.  u.  i.  touches  every  other  curve  of  the  family.  Since,  in  gen- 
eral, each  point  of  l.  u.  i.  may  be  approached  in  the  manner  indicated 
in  this  article,  the  above  theorem  may  be  thus  supplemented:  In 
general,  l.u.i.  is  touched  at  each' of  its  points  by  some  member  of 
the  family. 

Note  1.    The  family  of  circles,  Fig.  92,  will  serve  to  illustrate  this  theorem. 

Note  2.    An  analytical  proof  oi  the  theorem  is  given  in  Art.  157,  Note  3. 

Note  3.  It  is  necessary  to  use  the  qualifying  phrase  in  general  in  the 
enunciation  of  the  theorem,  for  there  are  some  families  of  curves  (viz.  curves 
having  double  points  and  cusps,  see  Arts.  163,  164),  in  which  a  part  of  I.  n.  i. 
may  not  touch  any  member  of  the  family.  It  is  beyond  the  scope  of  this 
book  to  go  into  these  cases  in  detail,  (See  Edwards,  Treatise  on  the  Biff.  Cnl., 
Art.  365  ;  Murray,  Differential  Equations,  Chap.  IV.)  Illustrations  may  be 
obtained  by  sketching  some  curves  of  the  families  {y  -f-  c)"^  =  x^  and 
(ij  4-  c)2  =  x{x  -  3)2. 


156,  157.]  ENVELOPES.  281 

157.  To  find  the  envelope  of  a  family  of  curves  having  one  pa- 
rameter. It  is  in  accordance  with  the  definitions  and  theorem 
in  Arts.  154-156  to  say  that  the  envelope  of  a  family  of  curves 
fix,  y,  a)  =  0,  if  there  he  an  envelope,  is,  in  general,  the  locus  of  the 
limiting  position  of  the  intersection  of  any  one  of  the  curves  of  the 

/am%,  say  the  curve 

f(x,  y,  a)  =  0  (1) 

with  another  curve  of  the  family,  viz. 

/(a^,2/,«  +  Aa)  =  0  (2) 

when  the  second  cwve  approaches  coincidence  with  the  first;  that 
is,  when  Aa  approaches  zero. 

From  (1)  and  (2),  f{x,  y,a  +  ^a)-f(x,  y,a)  =  0; 

hence  f(x,y,a  + ^a)-f(x,y,  a)  ^^^ 

Aa  ^  ^ 

Now  Equations  (1)  and  (3)  may  be  used,  instead  of  (1)  and  (2), 
to  find  the  points  of  intersection  of  curves  (1)  and  (2).  If  Aa  =  0, 
the  point  of  intersection  approaches  an  ultimate  point  of  inter- 
section.    When  (Arts.  22,  79)  Aa  =  0,  Equation  (3)  becomes 

£f{x,y,a)=0.  (4) 

Thus  the  coordinates  x  and  y  of  the  point  of  ultimate  inter- 
section of  curves  (1)  and  (2)  satisfy  Equations  (1)  and  (4);  and, 
accordingly,  satisfy  the  relation  which  is  deduced  from  (1)  and 
(4)  by  the  elimination  of  a.  Hence,  in  order  to  find  the  equation 
of  I.  u.  i.  of  the  family  of  curves  f{x,  y,  a)  =  0  eliminate  a  betiveen 
the  equations 

f(x,  y,a)=0  and  ^  f(x,  y,  a)  =  0.  (5) 

The  result  obtained  is,  in  general,  also  the  equation  of  the 
envelope. 

Note  1.  A  slightly  different  way  of  making  the  above  deduction  is  as 
follows.     Let  the  equations  of  two  curves  of  the  family  be 

f(x,  y,a)  =  0    (6),  and         f(x,  y,a-\-h)  =  0.  (7) 


282  INFINITESIMAL   CALCULUS.  [Ch.  XVIII. 

By  Art.  64,  Eq.  (3),  Equation  (7)  may  be  written 

/(ic,  y,  a)  +  h^  f{x,  y,  a -j-  eh)  =  0,  in  which  |  ^  |<  1.  (8) 

•J 
By  virtue  of  (6)  this  becomes  -^f(x,  2/?  «  +  ^h)  =  0.  (9) 

Accordingly,  the  coordinates  of  the  intersection  of  curves  (6)  and  (7) 
satisfy  (6)  and  (9).  When  h  becomes  zero,  the  point  of  intersection  becomes 
an  ultimate  point  of  intersection.     Hence  the  ultimate  points  of  intersection 

satisfy  equations  /(x,  y,  a)  =  0  and  ^/(x,  y,  a)  =  0,  and,  accordingly,  the 
a-eliminant  of  these  equations.* 

Note  2.  For  an  interesting  and  useful  derivation  of  result  (5)  for  cases 
in  which  /(x,  y,  a)  is  a  rational  integral  function  of  a,  see  Lamb's  Calculus, 
Art.  157. 

Note  3.  To  show  that,  in  general,  the  a-eliminant  of  Equations  (5)  touches 
any  curve  of  the  family. 

Let  the  second  of  Equations  (5)  on  being  solved  for  a  give  a  =  (f)(x,  y). 
Then  the  equation  of  the  I.  u.  i.  of  the  family  of  curves  /(x,  y,  a)  =  0  is 

/(x,  y,  «)  =  9  in  which  a  =  0(x,  y).  (10) 

civ 

The  slope  ~  of  any  one  of  the  family  of  curves  /(x,  y,  a)  =  0  is  given  (see 

Art.  56),  by  the  equation      '   ;)f      nf  //,, 

^  +  ^'^  =  0.  (11) 

dx     dy  dx  ^    ' 

The  slope  -^  of  the  I.  u.  i.  is  obtained  from  Equations  (10).     On  taking 

the  total  x-derivative  in  the  first  of  these  equations, 


But  by  the  second  of  (5), 


df.dfdydf  da 
dx'^  dy  dx^  da  dx~    ' 

(12) 

>),  -;;r-  =  0,  and  accordingly,  (12)  reduces  to 

df.dfdy 
dx  '^  dy  dx      ' 

(13) 

Thus  the  slope  of  the  I.  u.  i.  and  the  slope  of  any  member  of  the  family 
are  both  given  by  the  same  equation.  Hence,  at  a  point  common  to  any 
curve  and  the  I.  u.  i.,  the  slopes  of  both  are  the  same,  and  accordingly,  the 
curve  and  the  I.  u.  i.  touch  at  that  point. 

Sometimes  the  value  of  ~  obtained  from  (11)  is  indeterminate  in  form, 

dx  ^     ^ 

and  the  slopes  of  the  curve  and  I.  u.  i.  may  not  be  the  same.     See  Arts.  165, 
156  (Note  3),  and  Lamb,  Calculus,  Art.  158. 

*  This  method  of  finding  envelopes  appears  to  be  due  to  Leibnitz. 


157.]  ENVELOPES.  283 

EXAMPLES. 

1.  Find  the  envelope  of  the  family  of  circles  (see  Art.  154) 

{X  -  cY  +  y^  =  4.  (1) 

Here,  on  differentiation  with  respect  to  the  parameter  c, 

2  (X  -  c)  =  0.  (2) 

The  elimination  of  c  between  these  equations  gives 

y  =  4, 
which  represents  the  two  straight  lines  y  =  2,  y  =—2. 

2.  Find  the  envelope  of  the  family  of  lines 

y  =  mx  —  2  pm  —  pm^,  (1) 

in  which  m  is  the  parameter.  (This  is  the  equation  of  the  general  normal  of 
the  parabola  y'^  =  4ipx  ;  see  works  on  analytic  geometry.)  On  differentiation 
with  respect  to  the  parameter  m, 

0  =  x-2p -Spm^.  (2) 

The  m-eliminant  of  (1)  and  (2)  is  the  equation  of  the  envelope. 
On  taking  the  value  of  m  in  (2)  and  substituting  it  in  (1),  and  simplifying 
and  removing  the  radicals,  there  is  obtained 

27  py2=4(x-2  py.  (3) 

Note  4.  In  Art.  152  it  is  shown  that  the  normals  to  a  curve  touch  its 
evolute.  It  also  appears  from  Art.  152  that  each  tangent  to  an  evolute  is 
normal  to  the  original  curve.  Accordingly,  it  may  be  said  that  the  evolute 
of  a  curve  is  the  envelope  of  its  normals,  and  likewise  that  the  evolute  of  a 
curve  is  the  I.  u.  i.  of  its  {family  of)  normals.  (See  Art.  152,  Note  2,  and 
Art.  149,  Note  5.) 

Note  5.     Compare  Ex.  1,  Art.  151,  Ex.  2  above,  and  Ex.  1,  Art.  158. 

3.  If  A,  B,  C  are  functions  of  the  coordinates  of  a  point  and  m  a 
variable  parameter,  show  that  the  envelope  of  Ain^  +  Bin  +  C  —  0  is 
^2-4^0  =  0. 

Note  6.  The  result  in  Ex.  3  is  the  same  in  form  as  the  condition  that  the 
roots  of  the  quadratic  equation  in  m  he  equal.  This  result  is  immediately 
applicable  in  many  instances.  It  is  very  easily  deduced  on  taking  the  point 
of  view  explained  in  the  article  mentioned  in  Note  2. 

4.  Deduce  the  result  in  Ex.  3  without  reference  to  the  calculus. 
Apply  this  result  to  Ex.  1. 


284  INFINITESIMAL   CALCULUS.  [Ch.  XVIII. 

IV. B.     Make  figures  for  the  following  examples. 

5.    Find  the  curves  whose  tangents  have  the  following  general  equations, 
in  which  m  is  the  variable  parameter : 


(1)  y  =  mx  -\-  aVl  -\-  m^.  (2)  y  =  mix  +  y/a^hrir-  -\-  b'^. 

(3)  y  =  mx±  y/arii^  -\-  bm  +  c.  (4)  y  —  mx  +  a  Vm. 

(5)  mH  —  my-\-a.  (Q)  y  —  b  —  m{x  —  a)  +  rVl  +  w'-^. 

6.  Find  the  envelopes  of  the  following  lines  : 

(1)  x  sin  ^  —  y  cos  ^  +  a  =  0,  parameter  6.  (2)  x  -^  ysvnd  =  a  cos  0, 

parameter  d.       (3)  ax  sec  a  —  by  cosec  cj  =  a^  _  52^  parameter  «. 

7.  Find  the  envelopes  of  (1)  the  parabolas  ?/2  =  4  a{x  —  a),  parameter  a  ; 
(2)  the  parabolas  cy'^  =  a^(x  —  a),  parameter  a. 

8.  Show  that  if  A,  B,  C  are  functions  of  the  coordinates  of  a  point,  and 
a  a  variable  parameter,  the  envelope  of  A  cos  ct  +  ^  sin  a  =  C  is  ^2  _|_  ^2  _  (j%^ 

9.  Find  the  evolute  of  the  ellipse  x  =  a  cos  (p,  y  ==  b  sin  <p,  considering 
the  evolute  of  a  curve  as  the  envelope  of  its  normals. 

10.  One  of  the  lines  about  a  right  angle  passes  through  a  fixed  point,  and 
the  vertex  of  the  angle  moves  along  a  fixed  straight  line  ;  find  the  envelope 
of  the  other  line. 

11.  From  a  fixed  point  on  the  circumference  of  a  circle,  chords  are 
drawn,  and  on  these  as  diameters  circles  are  described.  Show  that  they 
envelop  a  cardioid. 

158.  To  find  the  envelope  of  a  family  of  curves  having  two  parame- 
ters.    Let  ^/  ,v      f. 

fix,  y,  a,  b)  =  0 

be  a  family  of  curves  which  has  two  parameters.     If  there  is  a 
given  relation  between  these  parameters,  say 

F(a,  b)  =  0, 

then  the  two  parameters  practically  come  to  one,  and  accordingly, 
the  case  reduces  to  that  considered  in  Art.  157. 

EXAMPLES. 

1.  Find  the  envelope  of  the  normals  to  the  parabola  y^  =  4px.  The 
equation  of  the  normal  at  any  point  (xi,  yi)  on  this  parabola  is 

y-yii-P^(x-r-xO  =  o, 


158.]  ENVELOPES.  285 

This  reduces  to       2py  —  2  pyi  +  xyi  —  Xipi  =  0.  ,  •  (1) 

Here  there  are  two  parameters,  Xi  and  yi.    They  are  connected  by  the 
^^l^^i^^  y{^  =  ^px,. 

Hence  (1)  becomes     2py  —  2pyi-j-xyi  —  ^  =  0,  (2) 

4p 

which  involves  only  a  single  parameter  yi.  On  differentiating  in  (2)  with 
respect  to  the  parameter  yi  and  then  eliminating  yi,  there  will  appear  the 
equation  of  the  envelope,  viz. 

27  py^  =  A  (x- 2  p)^ 
Compare  Ex.  1  with  Ex.  1,  Art.  151,  and  Ex.  2,  Art.  157. 

Note.     This  problem  may  be  expressed :    Find  the  envelope  of  the  Hue 
(1),  given  that  the  point  (xi,  yi)  moves  along  the  parabola  y'^  =  4j9x. 

2.  Find  the  envelope  of  the  line 

-+y^i  (1) 

a     0 
when  the  sum  of  its  intercepts  on  the  axes  is  always  equal  to  a  constant  c. 
Since  a  +  b  =  c,  (2) 

Equation  (1)  may  be  written  -  -| ^  =  1, 

a     c  —  a 

i.e.  (c  —  a)x  +  ay  =  ac  —  a^.  (3) 

Thus  (1)  is  transformed  into  an  equation  involving  a  single  parameter  a. 
On  differentiating  in  (3)  with  respect  to  the  parameter  a, 

—  x  +  y  =  c  —  2a.  (4) 

The  elimination  of  a  between  (3)  and  (4)  gives 

x^-^y"^  +  c^  =  2cx  +  2xy  -\-2cy. 

This  reduces  to  Vx  -\-  Vy  =  Vc. 

See  Ex.  7,  Art.  59. 

The  elimination  of  a  and  b  can  also  be  performed  thus : 

Differentiation  in  (1)  and  (2)  with  respect  to  a  gives 

_.£_i^^  =  Oand  1+^  =  0. 
a^      b^  da  da 


On  equating  the  values  of  — , 
da 


^^  =  1;  whence  5  =  ^.  (6) 


286  INFINITESIMAL    CALCULUS.  [Ch.  XVIII. 

From  (2)  and  (5),    a=      f^^  _,    b  =  ~P^ — 

Vx  +  Vy  Vx  +  Vy 

On  substitution  in  (1)  and  reduction,  Vx  +  Vt/  —  Vc. 

Tliis  second  metliod  is  generally  more  useful  than  that  used  in  Ex.  1  and 
in  the  first  way  of  working  Ex.  2,  in  cases  when  the  two  parameters  are 
involved  symmetrically  in  the  equation  and  in  the  expression  of  the  relation 
between  the  parameters. 

3.  Find  the  envelope  of  the  straight  lines  the  product  of  whose  intercepts 
on  the  axes  of  coordinates  is  equal  to  a'. 

4.  Find  the  envelope  of  a  straight  line  of  fixed  length  a  which  moves  with 
its  extremities  in  two  lines  at  right  angles  to  each  other. 

5.  A  set  of  ellipses  which  have  a  common  centre  and  axes,  and  in 
which  the  sum  of  the  semi-axes  is  equal  to  a  constant  a,  is  drawn :  find  the 
envelope  of  the  ellipses. 

6.  Show  that  the  envelope  of  a  family  of  co-axial  ellipses  having  the 
same  area  consists  of  two  conjugate  rectangular  hyperbolas. 

7.  Circles  are  described  on  the  double  ordinates  of  the  parabola 
y'^  =  4  ax  as  diameters :  show  that  the  envelope  is  the  equal  parabola 
?/2  —  4a(x  +  a). 

8.  Circles  are  described  having  for  diameters  the  double  ordinates  of 
the  ellipse  whose  semi-axes  are  a  and  b  :  show  that  their  envelope  is  the 
co-axial  ellipse  whose  semi-axes  are  Va'^  -\-  b'^  and  b. 

9.  About  the  points  on  a  fixed  ellipse  as  centre,  ellipses  are  described 
having  axes  equal  and  parallel  to  the  axes  of  the  fixed  ellipse  :  show  that 
their  envelope  is  an  ellipse  whose  axes  are  double  those  of  the  fixed  ellipse. 

10.  A  straight  line  moves  so  that  the  sum  of  the  squares  of  the  perpen- 
diculars on  it  from  two  fixed  points  (±  c,  0)  is  constant  (=2  ^■'^)  :  show  that 

its  envelope  is  the  conic   — 1-  ^  =  1. 

11.  If  the  diiJerence  of  the  squares  in  Ex.  10  is  constant,  show  that  the 
envelope  is  a  parabola. 

12.  Show  that  if  the  corner  of  a  rectangular  piece  of  paper  be  folded 
down  so  that  the  sum  of  the  edges  left  unfolded  is  constant,  the  crease  will 
envelop  a  parabola. 

Asymptotes. 

159.  Rectilinear  asymptotes.  In  preceding  studies  acquaint- 
ance has  been  made  with  two  lines  related  to  the  hyperbola, 
called  asymptotes  and  possessing  the  following  properties : 
(a)  These  lines  are  the  limiting  positions  which  the  tangents  to 
the  hyperbola  approach  when  the  points  of  contact  recede  for  an 


159.]  ASYMPTOTES.  287 

infinite  distance  along  the  curve  (or,  as  it  may  be  expressed, 
recede  towards  infinity) ;  (b)  the  lines  themselves  do  not  lie 
altogether  at  infinity.  (This  is  the  mathematical  way  of  saying 
that  the  lines  run  ac^ross  the  field  of  view;  in  fact,  in  the  case  of 
the  hyperbola  they  pass  through  the  centre  of  the  curve.) 

Besides  hyperbolas  there  are  many  other  curves  which  have 
branches  extending  to  an  infinite  distance  and  which  have  associ- 
ated Avitli  them  certain  lines  having  properties  like  (a)  and  (6) ; 
namely,  lines :  (1)  that  are  the  limiting  positions  which  the  tan- 
gents to  the  infinite  branches  approach  when  the  points  of  contact 
recede  towards  infinity ;  (2)  that  do  not  lie  altogether  at  infinity ; 
for  instance,  using  rectangular  coordinates,  lines  that  pass  within 
a  finite  distance  of  the  origin. 

Lines  having  properties  (1)  and  (2)  are  called  asymptotes  of  the 
curves.  Thus  an  ellipse  cannot  have  an  asymptote,  since  it  has 
no  branch  extending  to  infinity  (see  Ex.  3,  Art.  161).  Again 
the  parabola  ?/-  =  4p.r  has  no  asymptote,  for  (see  Ex.  4,  Art.  161) 
the  tangent  at  an  infinitely  distant  point  of  this  parabola  crosses 
each  of  the  axes  of  coordinates  at  an  infinite  distance  from  the 
origin,  and,  accordingly,  no  part  of  this  tangent  can  be  in  sight ; 
i.e.  it  lies  wholly  at  infinity.  (The  asymptotes  are  apparent  in 
the  figures  on  pages  410-414.) 

It  will  now  be  shown  how  an  examination  may  be  made  for  the 
asymptotes  of  curves  whose  equations  have  the  form 

F{x,  y)  =  0,  (1) 

where  F{x,  ?/)  is  a  rational  integral  function  of  x  and  y.  For  this 
it  is  necessary  to  call  to  mind  the  algebraic  property  stated  in  the 
following  note. 

Algebraic  Note.     On  substituting  ~  for  x  in  the  rational  integral  equation 

Coa:«  +  cix»-i  +  c^x^--  +  •••  +  c„_ix  +  c„  =  0,  (a) 

and  clearing  of  fractions,  it  becomes 

Co  +  Cit  +  czt^  +  •••  +  c„_i«"-i  +  CnV'  =  0.  (5) 

It  is  shown  in  algebra  that  if  a  root  of  Equation  (b)  approaches  zero,  Cq 
approaches  zero  ;  and  that  if  a  second  root  also  approaches  zero,  ci  also 

approaches  zero.     But,  since  x  =    ,.  when  a  root  of  (&)  approaches  zero,  a 


288  INFINITESIMAL    CALCULUS.  [Ch.  XVIII. 

root  of  (a)  increases  beyond  all  bounds,  i.e.,  to  use  a  common  phrase,  it 
approaches  infinity.  Hence,  the  condition  that  a  root  of  (a)  approach 
infinity  is  that  cq  approach  zero,  and  the  condition  that  a  second  root  of  (a) 
at  the  same  time  approach  infinity  is  that  Ci  also  approach  zero ;  and  so  on 
for  other  roots  approaching  infinity.  This  is  briefly  expressed  by  saying  that 
equation  (a)  has  a  root  equal  to  infinity  when  cq  =  0,  and  has  two  roots 
equal  to  infinity  when  co  =  0  and  ci  =  0. 

160.  To  find  asymptotes  which  are  parallel  to  the  axes  of  coordi- 
nates. Suppose  that  the  equation  of  the  curve  F(x,  y)  =  0  [Art. 
159  (1)]  is  of  the  7?th  degree,  and  that  the  terms  in  the  first  mem- 
ber of  this  equation  are  arranged  according  to  decreasing  powers 
of  y.     Then  the  equation  has  the  form 

i^o2/"  +  i^i2/"~^  +  P-2y"~-  H h  Pn-iV  -\-Pn  =  0.  (1) 

Here,  po  is  a  constant ;  2h  i^3,y  be  an  expression  in  x  of  the  first 
degree  at  most,  say  ax-\-b;  p.2  may  be  of  the  second  degree  at 
most,  say  cx^  -{-  dx -{-  e -,  jh  i^^J  be  of  the  third  degree  in  x  at 
most;  •••;  and  p^  may  be  of  the  ?ith  degree  in  x  at  most.  For 
if  any  one  of  tlie  respective  p's  were  of  a  higher  degree  than  that 
specified  above,  F{x,  y)  would  be  of  a  higher  degree  than  the  nth. 

Ex.  1.  Arrange  the  first  members  of  the  following  equations  (a)  in 
descending  powers  of  x  ;  (h)  in  descending  powers  of  y  : 

'      (1)  xy  -  ay  -bx  =  0.  (2)  x^  +  xi/^  -\- 2x'^  -  2y^  -  7  x  +  iy  -  11  =  0. 

(3)  2xy^-  x^y  +  3y'^  -Sx^  i-ixy  -2x-\-l  y  +  1  =0. 

(4)  y^  -\-  x^y  -{- x"^  +  2  xy  -\- 7  X -\- 2  =  0. 

Now  suppose  that  in  (1)  Pq  =  0  ;  then  (1)  may  be  written 

0  . 2/"  +  (ax  +  ^)?/"~^  +  (cx^  +  dx  +  e)^/""^  +2hy"~^  H 

-hPn-iy-\-Pn  =  0.  (2) 

If  this  be  regarded  as  an  equation  of  the  nth  degree  in  y,  then 
to  any  finite  value  of  x  there  correspond  n  values  of  y,  one  of 

which  is  iniinitely  great.     If  also  ax  -\-b  =  0,  i.e.  if  x  = ,  a 

second  of  the  n  values  of  y  is  infinitely  great.  In  a  similar  way 
points  whose  abscissas  are  infinitely  great  and  whose  ordinates  are 
finite  may  be  found. 


160.]  ASYMPTOTES.  289 

Ex.  2.  Thus  in  Ex.  1  (1)  the  equation,  which  is  of  the  second  degree,  may 
be  written  y(x  —  a)  —  bx  =  0.  Accordingly  one  value  of  y  is  infinite  ;  a  second 
value  of  y  is  infinite  when  x  =  a. 

Ex.  3.    Show  that  a  second  value  of  x  is  infinite  when  y  =  b. 

It  will  now  be  shown  that  an  infinite  ordinate  ivhose  distance 
from  the  origin  is  finite  is  tangent  to  the  curve  at  the  infinitely  dis- 
tant point. 

On  differentiating  in  (2)  with  respect  to  x  and  solving  for  -^, 

dx 

dy  ^  ay»-^  +  (2  ex  +  d)y»-^  +  •••  +  p'n 3 

dx         (n  -  l){ax  +  b)y"-'-h(n-2){cx2-\-dx  +  e)y»-''^-\-  -■ -\- Pn~i     ^^ 

When  X  =  — ,  the  numerator  in  the  second  member  is  an  infinity  of  an 

a 
order  at  least  two  higher  than  the  denominator,  and  hence  the  value  of  the 

fraction  is  then  infinite.     Hence  the  line  x  =  —  is  a  tangent  at  any  point 

b  ^ 

for  which  x  =  —  and  y  =  ao. 

a 
In  a  similar  way  it  can  be  shown  tfiat  if  one  of  the  values  of  x  in  Equa- 
tion (1),  Art.  159,  is  infinite  when  y  =  c,  in  which  c  is  finite,  then  y  =  c  is 
a  tangent  at  any  point  for  which  x  =  <x>  and  y  =  c. 

Note  1.     If  [see  Eq.   (2)]  x  =  —   also  satisfies  cx^  -\-  dx  +  e  =  0,  then 

a 
three  values  of  y  in  F(x,  y)  =  0  are  infinitely  great  for  this  value  of  x.     The 

line  X  = is  then  an  inflexional  tangent  (see  Art.  78,  Note  1)  at  infinity. 

Note  2.  This  method  of  finding  asymptotes  parallel  to  the  axes  can  be 
applied  to  curves  whose  equations  are  not  of  the  kind  considered  above. 
Instances  are  given  in  Exs.  7,  8  (6),  (9)  that  follow. 

EXAMPLES. 

4.  Find  the  asymptotes  of  the  curves  in  Ex.  1. 

5.  Determine  the  finite  points  (if  they  exist)  in  which  each  asymptote 
in  Ex.  4  meets  the  curve  to  which  it  belongs. 

6.  Show  that  the  line  x  =  a  is  an  asymptote  of  the  curve  y  =  ^1±L 
when  0(a)  and  <p'{a)  are  finite.  ^  —  a 

Here,  Wm.^^y  =  oo.    Also  ^  =  (x- a)0^(x)- 0(x)  .  ^^^^^^  M^^.jk  =  ^. 

dx  (x  —  ay  dx 

Hence  x  =  a  is  a  tangent  at  an  infinitely  distant  point  (x  =  a,  ?/  =  oo). 

7.  Examine  y  =  tan  x  for  asymptotes. 

Here  y  =  +  oo  when  x  =  ^,  ^,  ^,  ....  ' 
^      ^  2'    2  '    2  ' 

Also,  ^  =  sec2.x.     Hence  ^  =  oo  when  x  =  '^,  "^,  ^,  .... 
dx  dx  2      2       2 

/.  X  =  -,  X  =  ^,  X  =  -^,  •••,  are  asymptotes. 
^  A  A 


290  INFINITESIMAL    CALCULUS.  [Ch.  XVIII. 

8.    Determine  the  asymptotes  of  the  following  curves  :    (1)  The  hyper- 

3  ^-.-    -. 8a-^ 


bola  xy  =  a^.     (2)  The  cissoid  y^  =  — -^ (3)  The  witch  y 

"la  —  X  a;2  +  4  a"^ 

(4)  (x2  -  a2)  (^2  _  ^,2)  ^  ^2^2.    (5>)  ^2^  ^  j/C-K  -  a)2.     (6)  J/  =  log  X.     (7)  y  =  e\ 
(8)  The  probability  curve  y  =  e--*^     (9)  ?/  =:  sec  x. 


161.  Oblique  asymptotes.  There  are  asymptotes  which  are  not 
parallel  to  either  axis.  The  method  of  finding  them  can  best  be 
shown  by  an  example. 

EXAMPLES. 

1.  Find  the  asymptotes  of  the  folium  of  Descartes  (see  page  413) 

x^  +  2/3  =  3  a  x?/.  (1) 

First  find  the  intersections  of  this  curve  and  the  line 

y  =  mx  +  h.  C23 

On  solving  these  equations  simultaneously, 

(1  +  w3)x3  +  3  {m%  -  am)x'^  +  3  (m&2  _  ab)x  +  b^  =  0. 

Line  (2)  is  a  tangent  to  the  curve  (1)  at  an  infinitely  distant  point,  if  two 
roots  of  this  equation  are  infinitely  great.     That  is,  if 

1  4-  m3  =  0,  and  m%  -  am  =  0.  (3) 

That  is,  on  solving  Equations  (3)  for  m  and  &,  if 

TO  =:  —  1,  and  b  =—  a. 

Hence,  the  asymptote  is  y  -^  x  -\-  a  =  0. 

Note  1.  A  curve  whose  equation  is  of  the  nth  degree  has  n  asymptotes, 
real  or  imagifiary.  This  may  be  apparent  from  the  preceding  discussion. 
For  proof  of  this  theorem  see  references  for  collateral  reading,  Art.  162. 

In  Ex.  1  two  values  of  7n  in  Equations  (3)  are  imaginary  ;  thus  curve  (1) 
has  one  real  and  two  imaginary  asymptotes. 

2.  Find  the  asymptotes  of  the  hyperbola  b^x^  —  a^y"^  =  a%^. 

3.  Show  by  the  method  used  in  Ex.  1  that  the  ellipse  b'^x'^  +  a'^y'^  =  a%'^ 
has  no  real  asymptotes. 

4.  Show  by  the  method  used  in  Ex.  1  that  the  parabola  y^  =  ipx 
does  not  have  an  asymptote. 


161.  J  ASYMPTOTES.  291 

5.  Find  the  asymptotes  of  the  following  curves :  (1)  y^  =  x^  -\-  x. 
(2)  X*  -  y*  -  3  x3  -  a;j/2  -  2  X  +  1  =  0.  (3)  xy{y  -  x)  =  3  x^  +  2  y2. 
(4)   (x2  -y2)2  _  4  2/2  ^  y  ^  2  X  +  3  =  0.     (5)  a;3  -  8  «/^  +  3  x-2  -  x?/  -  2  ?/2  =:  0. 

Note  2.    Other  methods  of  finding  asymptotes. 

a.  Find  the  values  of  the  intercepts  on  the  axes  of  coordinates  of  the 
tangent  at  a  point  (x',  y')  on  a  curve  [see  Art.  59,  Note  3  (1)],  when 
x'  =  oo,  or  y'  =  CO,  or  both  x'  and  y'  are  infinitely  great.  If  one  or  both  of 
these  intercepts  is  finite,  the  tangent  is  an  asymptote.  Its  equation  can  be 
written  on  finding  its  intercepts. 

6.  Apply  this  method  to  Exs.  2,  4,  above. 

h»  Find  the  length  of  the  perpendicular  from  the  origin  to  the  tangent 
at  (x',  y')  when  x'  =  oo,  or  «/'  =  oo,  or  both  x'  and  y'  are  infinitely  great. 
If  this  length  is  finite,  the  tangent  is  an  asymptote. 

7.  Do  Exs,  2,  4,  by  this  method. 

c.  By  means  of  the  equation  of  the  curve  express  y  in  terms  of  a  series 
in  decreasing  powers  of  x,  or  express  x  in  terms  of  a  series  in  decreasing 
powers  of  y.  From  one  of  these  expressions  there  may  sometimes  be  de- 
duced the  equation  of  a  straight  line  which,  for  infinitely  distant  points, 
closely  approximates  to  the  equation  of  the  curve. 

8.  Thus,  in  the  hyperbola  in  Ex.  2, 


hxf^  _a2\i 
a  \        x2J 


«V        2x^  /  a        x4x3 


ha^  ^ 


It  is  apparent  from  this  that  the   farther  away  the  points  on  the  lines 

y  =±—  are  taken,  the  more  nearly  will  they  satisfy  the  equation  of  the 
a 

hyperbola,  and  that  when  x  increases  beyond  all  bounds,  the  points  on  these 

lines  satisfy  the  equation  of  the  hyperbola.     Accordingly,  these  lines  are 

asymptotes. 

Note  3.  Curvilinear  asymptotes.  Expansioji  may  sometimes  reveal 
the  equation  of  a  curve  of  higher  degree  than  the  first  whose  infinitely  distant 
points  also  satisfy  the  equation  of  the  given  curve.  Accordingly  the  two 
curves  coincide  at  infinitely  distant  points.  The  two  curves  are  said  to  be 
asymptotic,  and  the  new  curve  is  called  a  curvilinear  asymptote  of  the 
original  curve.  For  a  discussion  on  curvilinear  asymptotes  see  Frost's  Curve 
Tracing,  Chaps.  VII.  and  VIII. 


292  INFINITESIMAL    CALCULUS.  [Ch.  XVIII. 

162.  Rectilinear  asymptotes :  polar  coordinates.  In  order  to  find 
the  asymptotes  of  the  curve     „,     ^.      ^ 

a  method  similar  to  ^that  outlined  in  Art.  161,  Note  2  (6),  can  be 

used.  First  find  the  value  of  0 
in  Equation  (1)  for  which  the 
radius  vector  ?*  is  infinitely  great. 
Suppose  that  this  value  of  0  is 
$1.  Thus  the  point  (r=co,  $=di) 
is  an  infinitely  distant  point  of 
the  curve.  If  the  tangent  TN  at 
this  infinitely  distant  point  is 
an  asymptote,  it  passes  within 
a  finite  distance  from  0.  Accord- 
^^^'  ^^'  ingly?  TN  is  parallel  to  the  radius 

vector,  and  the  subtangent  OM,  viz.  r^ —  (Art.  61)  is  finite  for 

EXAMPLES. 

1.  Find  and  draw  the  asymptote  to  the  reciprocal  spiral  rd  =  a. 

Here  r  =  -•     .-.  r  =  oo  when  ^  =  0. 

d 

Also  e-^'     ...  ^  =  _  4.     ...  r2  ^  =  -  a.        (See  Fig. ,  page 

r  dr         r2  dr  4^4  . 

Hence  the  asymptote  is  parallel  to  the  initial  line  and  at  a  distance  a  to 
the  left  of  one  who  is  looking  along  the  initial  line  in  the  positive  direction. 

Note  1.  The  convention  used  in  Ex.  1  is  as  follows  :  A  positive  subtan- 
gent is  measured  to  the  right  of  a  person  who  may  be  looking  along  the 
infinite  radius  vector  in  its  positive  direction,  and  a  negative  subtangent  is 
measured  toward  the  left. 

2.  Find  and  draw  the  asymptotes  to  the  following  curves:    (1)  rsin^ 

Q 

=  ad.     (2)  r  cos  0  =  a  cos  2  d.     63)  r  sin  -  =  a. 

Note  2.  Circular  asymptotes.  If  the  radius  vector  r  approaches  a  fixed 
limit,  a  say,  when  6  increases  beyond  all  bounds,  then  as  d  increases,  the  curve 
approaches  nearer  to  coincidence  with  the  circle  whose  centre  is  at  the  pole 
and  whose  radius  is  a.  This  circle,  whose  equation  is  r  =  a,  is  said  to  be  a 
circular  asymptote,  or  the  asymptotic  circle^  of  the  curve. 


162-161.]  SINGULAR   POINTS.  293 

3.    In  the   reciprocal  spiral,  Ex.   1,  if  ^  =  oo,  then    r  =  0.      Hence   the 
asymptotic  circle  is  a  circle  of  zero  radius,  viz.  the  pole. 

e 


4.   Find  the  rectilinear  and  the  circular  asymptote  of  r 


1 


References  for  collateral  reading  on  asymptotes.  McMahon  and 
Snyder's  Diff.  Cal.,  Chap.  XIV.,  pages  221-242  ;  F.  G.  Taylor's  Calculus, 
Chap.  XVI.,  pages  228-249,  and  Edwards's  Treatise  on  the  Differential  Cal- 
culus, Chap.  VIII.,  pages  182-210,  contain  interesting  discussions  on  asymp- 
totes, with  many  illustrative  examples.  For  a  more  extended  account  of 
asymptotes  see  Frost's  Curve  Tracing,  Chaps.  VI. -VIII.,  pages  76-129. 


Singular  Points. 

163.  Singular  points.  On  some  curves  there  are  particular 
points  at  which  the  curves  have  certain  peculiar  properties  which 
they  do  not  possess  at  their  points  in  general.  For  instance,  there 
are  points  of  maximum  or  minimum  ordinates  (Art.  75),  points  of 
inflexion  (Art.  78),  and  points  of  undulation  (Art.  78).  There  are 
also  points  through  which  a  curve  passes  twice  or  more  than  twice 
(see  Figs.  96  a,  b,  c),  and  at  which  it  has  two  or  more  different 
tangents ;  there  are  points  through  which  pass  two  branches  of  a 
curve  that  have  a  common  tangent  (Figs.  97  a,  h,  c,  d) ;  and  there 
are  other  peculiar  points  hereafter  described.  Points  of  maximum 
and  minimum  ordinates  depend  on  the  relative  position  of  a  curve 
and  the  axes  of  coordinates ;  the  peculiarities  at  the  other  points 
referred  to  above  are  independent  of  the  axes  and  belong  to  the 
curve  whatever  be  its  situation.  Points  at  which  a  curve  has 
peculiarities  of  this  kind  are  called  singular  points.  Some  of  these 
singular  points  are  considered  in  Arts.  164,  165. 

164.  Multiple  points.  Double  points.  Cusps.  Isolated  points. 
Multiple  points  are  those  through  which  a  point  moving  along  the 
curve,  while  changing  the  direction  of  its  motion  continuously, 
can  pass  two  or  more  times,  and  at  which  the  curve  may  have  two 
or  more  different  tangents. 

For  example,  in  moving  from  L  to  M  along  the  curves  in  Figs. 
96  a,  6,  c,  a  point  passes  through  A  and  C  three  times  and  through 
B  and  D  twice.  At  A  there  are  three  different  tangents,  at  C 
there  are  three,  and  at  B  and  D  there  are  two  each.    Points,  such 


294 


INFINITESIMAL   CALCULUS. 


[Ch.  XVIII. 


as  B  and  D,  througli  which  the  point  moving  along  the  curve, 
while  continuously  changing  the  direction  of  its  motion,  can  pass 


Fig.  96  a. 


Fig.  96  b. 


Fig.  96  c. 


tivice,  are  called  double  j^oints;  points  such  as  A  and  C  are  called 
triple  points.  The  curve  r  =  a  sin  2  ^  (see  p.  414)  has  a  quadruple 
point. 

Note  1.     Multiple  points  are  also  called  nodes.     (Latin  nodus,  a  knot.) 

Cusps  are  points  where  two  branches  of  the  curve  have  the  same 
tangent.     See  Figs.  97  a,  b,  c,  d. 

In  Fig.  97  a  both  branches  of  the  curve  stop  at  A  and  lie  on 
opposite  sides  of  their  common  tangent  at  A.  In  Fig.  97  b  both 
branches  stop  at  B  and  lie  on  the  same  side  of  the  tangent  at  B. 
Both  branches  of  the  curve  pass  through  C.  Accordingly  C  is 
sometimes  called  a  double  cusp.  If  a  point  is  moving  along  a 
curve  LKM  which  has  a  single  cusp  at  K  (Fig.  97  d),  there  is  an 


A  B 

Fig.  97  a.  Fig.  97  &. 


Fig.  97  c. 


Fig.  97  d. 


abrupt  (or  discontinuous)  change  made  in  the  direction  of  its 
motion  on  its  passing  through  K.  On  arriving  at  K  from  L  the 
moving  point  is  going  in  the  direction  a ;  on  leaving  K  for  M  the 
moving  point  is  going  in  the  direction  b.  Thus  at  K  it  has 
suddenly  changed  the  direction  of  its  motion  by  the  angle  ir. 

Note  2.  A  cusp  such  as  K  (Fig.  97  d)  may  be  supposed  to  be  the  final 
(or  limiting)  condition  of  a  double  point  like  D  (Fig.  06  c)  when  the  loop 
DB  dwindles  to  zero  and  the  two  tangents  at  D  become  coincident. 


164.]  SINGULAR  POINTS.  295 

Isolated  or  conjugate  points  are  individual  points  which  satisfy 
the  equation  of  the  curve  but  which  are  isolated  from  (i.e.  at  a 
finite  distance  from)  all  other  points  satisfying  the  equation. 

EXAMPLES. 

1.  Sketch  the  curve  y"  —  (x  —  a)(x  —  b)(x  —  c),  in  which  a,  6,  and  c, 
are  positive  and  a<^b<Cc. 

2.  Sketch  the  curve  y^  =  {x  —  a){x  —  b)^,  in  which  a<6  and  both 
are  positive. 

3.  Sketch  the  curve  y^  =  (x  —  a)'^{x  —  6),  in  which  a  and  b  are  as  in  Ex.  2. 

4.  Sketch  the  curve  y^  =  (x  —  a)=^  in  which  a  is  positive. 

The  sketcli  in  Ex.  1  will  show  an  oval  from  x  =  a  to  x  =  6,  a  blank  space 
from  x=b  to  x  =  c^  and  a  curve  extending  from  x=c  to  the  right.  The  sketch 
in  Ex.  2  will  show  a  curve  having  a  double  point  at  (6,  0).  The  sketch  in 
Ex.  3  will  show  a  conjugate  point  at  (a,  0),  a  blank  space  from  x  =  a  to  x  =  b, 
and  a  curve  extending  from  x=b  to  the  right.  The  sketch  in  Ex.  4  will  show 
a  curve  having  a  cusp  at  (a,  0). 

Note  .3.  Other  singular  points.  There  also  are  points  called  salient 
points,  like  D  (Fig.  98),  for  instance,  where  two  branches  of  the  curve  stop 
but  do  not  have  a  common  tangent.  In  these 
cases  the  slope  of  the  tangent  changes  abruptly. 
Accordingly,  \i  y  =  (p(x)  be  the  equation  of  the 
curve,  (f>'(x)  is  discontinuous  at  the  salient 
points.  (See  Exs.  5,  6,  below.)  A  salient  point 
such  as  D  may  be  considered  to  be  the  limiting  condition  of  a  double 
point  like  D  (Fig.  96  c),  when  the  loop  Dli  dwindles  to  zero  but  the  two 
tangents  at  D  do  not  become  coincident.  (Compare 
"A    Note  2.) 

There  are  also  stop  points,  as  A,  Fig.  99,  where  the 
Fig.  99.  curve  stops  and  has  but  one  branch.     See  Ex.  7. 

1 

5.  In  the  curve  y(l  -{-  e')  =  x  show  that  when  x  approaches  the  origin 
from  the  positive  side,  the  slope  is  zero  ;  if  from  the  negative  side,  the  slope 
is  1.    The  origin  is  thus  a  salient  point.        Suggestion  :    The  slope  at  the 

origiH  may  be  taken  as  linix^o  -•  |     Find  the  angle  between  the  branches  at  the 
origin.  ^  -^ 

6.  In   the  curve  y  =  x- show  that  when  x  approaches  the  origin 

e^+1 
from  the  positive  side  the  slope  is  -|-  1,  and  if  from  the  negative  side,  the 
slope  is  —  1.     The  origin  thus  is  a  salient  point :  find  the  angle  between  thq 
branches  there. 

7.  Show  that  the  origin  is  »  gtop  point  in  the  curve  ?/  ^  aj  log  x. 


296  INFINITESIMAL   CALCULUS.  [Ch.  XVIIL 

165.  To  find  multiple  points,  cusps,  and  isolated  points.  From 
Art.  164  it  is  evident  that  in  order  to  determine  the  character  of 
a  point  on  a  curve,  it  is  first  of  all  necessary  to  examine  the  tan- 
gent (or  tangents)  there.     Let  the  equation  of  the  curve  be 

/(^,2/)  =  0,  ■  (1) 

and  let  f{x,  y)  be  a  rational  integral  function  of  x  and  y.     Then 

Sf 

|  =  -|.     [Art.  84,  (4).]  (2) 

dy 

Kow  at  a  multiple  point  or  a  cusp  -^  has  not  a  single  definite 

ux 

value,  and,  accordingly,  at   such  points   —  in  (2)  must  have  an 

indefinite  form,  viz.  the  form  —*     Hence,  at  a  multiple  point  of 

curve  (1)  ^ 

^=0  and  ^  =  0.  (3) 

ax  dy 

The  solutions  of  Equations  (3)  will  indicate  the  points  which  it 
is  necessary  to  examine,  t     At  these  points 


dx     0' 


(4) 


the  indefinite  form  in  the  second  member  can  be  evaluated  by  the 
method  explained  in  the  Appendix,  Note  C,  and  applied  in  Note 
below. :|:    Suppose  that  the  second  member  of  (4)  has  been  evaluated 

and  the  resulting  equation  solved  for  —  •     Then:    If  —  has  two 

(iX  C(X 

real  and  different  values  at  the  point  under  consideration,  the 
point  is  a  double  point  or  a  salient  point;  if  —  has  three  real 

and  different  values  there,  it  is  a  triple  point ;  and  so  on.     If  -^ 

cix 

*  This  is  frequently  called  an  '■'■  indeterminate,''''  form.  The  evaluation  of 
(so-called)  "  indeterminate  forms  ^'  is  discussed  in  the  Appendix,  Note  C. 

t  The  values  of  x  and  y  that  satisfy  Equations  (3),  may  give  points  that  are 
not  on  the  curve.     Of  course  these  points  need  not  be  examinecl  further. 

J  Or  by  othef  metl^od^  referred  to  in  Appendix,  I^ote  C, 


165.] 


SINGULAR  POINTS. 


297 


has  two  real  and  equal  values  at  the  point  which  is  being  examined 
the  point  is  a  cusp,     ^f  ^ 
is  an  isolated  point 


If  ^^  has  imaginary  values  at  the  point,  it 
ctx 


If  the  point  is  a  cusp,  the  kind  of  cusp  can  be  found  by  further  examina- 
tion of  the  curve  in  the  neighborhood  of 
the  point.  For  example,  if  (xi,  yi)  is 
known  to  be  a  cusp  and  it  is  found  that 
for  X  =  Xi  —  h  (h  being  infinitesimal),  y  is 
imaginary,  then  the  curve  does  not  extend 
tlirough  (a;i,  yi)  to  the  left,  and  thus  the 
cusp  is  not  a  double  cusp.  If  for  x=xi  +  h, 
the  value  of  the  ordinate  of  the  tangent  at 
(a^i,  yi)  is  less  than  the  ordinates  of  both 
branches  of  the  curve,  the  cusp  is  as  in  Fig. 
100.  In  a  similar  way  tests  may  be  devised 
and  applied  in  special  cases  as  they  arise. 

Note.     The  evaluation  of  the  second  member  of  Equation  (2)  gives, 
by  Appendix,  Note  C,  and  Art.  81,  (5) 


dy 
dx 


d^f  ,     ay    dy 

dx'^     dy  dx  dx 

dV       dVdy 

dx  dy    dy^  dx 


(5) 


If  the  second  member  of  (5)  is  not  indefinite  in  form,  this  equation,  on 
clearing  of  fractions  and  combining,  becomes 


dV(dy\ 
dy^  \dxl 

dy 


dydx  dx     dx^ 


(6) 


a  quadratic  equation  in  ^.     By  the  theory  of  quadratic  equations,  the  two 

dv 
values  of  ^  are  real  and  different,  real  and  equal,  or  imaginary,  according  as 

/  d'^f  V  dH    dH 

(  ^-^  j  is  respectively  greater  than,  equal  to,  or  less  than  V^  •  V^- 

the  point  is  a  double  point,  a  cusp,  or  a  conjugate  point,  according  as 


Hence, 


/  d'\f 
\dydx 


>, 


or  < 


5^    dV 
dy'  dx''' 


If  the  second  member  of  (5)  also  is  indefinite  in  form,  proceed  as  required 
by  Note  C,  remembering  that  ~  here  is  constant.     The  resulting  equation 


will  be  of  the  third  degree  in 


dy 
dx 


298  INFINITESIMAL   CALCULUS.  [Ch.  XVIII. 


EXAMPLES. 

1.  Examine  the  curve  a;^  -  ?/2  _  7  x-  +  4  ?/  +  15  x  —  13  =  0  for  singular 
points. 

Here  d|/^  _  Sx^  -  14x  +  15^ 

dx  -22/4-4  ^  ^ 

On  giving  each  member  the  indefinite  form  -,  and  solving  the  equations 

3  x2  -  14  X  +  15  =  0, 

-2?/ +  4  =  0, 

it  results  that  x  =  3  or  |,  and  y  =  2. 

Substitution  in  the  equation  of  the  curve  shows  that  x  =  f ,  ?/  =  2,  do  not 
satisfy  the  equation,  and  that  x  =  3,  y  =  2  do.  Accordingly,  the  point  (3,  2) 
is  the  point  to  be  further  examined. 

On  evaluating,  by  the  method  shown  in  the  Appendix,  the  second  member 
of  (1)  for  the  values  a;  =  3,  y  =  2,  it  is  found  that 

dy         6x  — 14       ,  I dy\^      -,         ,  dy 


dx  _2^ 

dx 


■  whence  [~\   =2,  and  -}-  =  ±  V2. 
\dxl  dx 


Thus  the  curve  has  a  double  point  at  (3,  2),  and  the  slopes  of  the  tangent 
there  are  4-  V2  and  —  \/2. 

[The  curve  consists  of  an  oval  between  the  points  (1,  2),  and  (3,  2),  and 
two  branches  extending  to  infinity  to  the  right  of  (3,  2).] 

2.  Sketch  the  curve  in  Ex.  1. 

3.  Examine  the  following  curves  for  singular  points  : 

(1)  a^y^  =  x-2(a2  _  x2).  (2)  x^  +  9  ^2  -  ^/'-^  +  27  x  +  2  y  +  26  =  0. 

(3)  ?/3  -  x2  -  3  ?/2  +  3  y  +  4  X  -  5  =  0.      (4)  The  curve  in  Ex.  "5  (5),  Art.  161. 

(5)  x3  +  2/3  +  3  x;hj  +  3  X2/2  -  10  2/^  -  16  xy  -  10  x2  +  25  x  +  29  2/  -  28  =  0. 

(6)  x3  -  2/2  -  10  x2  +  33  X  -  36  =  0. 

166.  Curve  tracing.  Some  of  the  matters  involved  in  curve 
tracing  have  been  discussed  in  Arts.  75-78,  159-165.  To  do  more 
than  this  is  beyond  the  scope  of  a  primary  text-book  on  the 
calculus.  The  topic  is  mentioned  here  merely  for  the  purpose  of 
giving  a  few  exercises  whose  solutions  require  the  simultaneous 
application  of  methods  for  finding  points  of  maximum  and  mini- 
mum, asymptotes,  and  singular  points. 


166.]  SINGULAR   POINTS.  ^99 

Note  1.  For  a  fuller  elementary  treatment  of  singular  points  and  curve 
tracing,  see  McMahon  and  Snyder,  Diff.  Cal.,  Chaps.  XVII.,  XVIII., 
pp.  275-306;  F.  G.  Taylor,  Calculus,  Chaps.  XVII.,  XVIII.,  pp.  250-278; 
Edwards,  Treatise  on  Diff.  Cal.,  Chaps.  IX.,  XII.,  XIII.;  Echols,  Calculus, 
Chaps.  XV.,  XXXI.,  pp.  147-164,  329-346.  The  classic  English  work  on  the 
subject  is  Frost's  Curve  Tracing  (Macmillan  &  Co.),  a  treatise  which  is 
highly  praised  both  from  the  theoretical  and  the  practical  point  of  view.* 

Note  2.  For  the  application  of  the  calculus  to  the  study  of  surfaces  (their 
tangent  lines  and  planes,  curvature,  envelopes,  etc.)  and  curves  in  space,  see 
Echols,  Calculus,  Chaps.  XXXII. -XXXV.,  pp.  347-390,  and  the  treatises  of 
W,  S.  Aldis  and  C.  Smith  on  Solid  Geometry. 

EXAMPLES. 

1.  Trace  the  curves  in  Ex.  8,  Art.  160;  in  Ex.  5,  Art.  161;  in  Ex.  2, 
Art.  162;  in  Ex.  3,  Art.  165. 

.    2.   Trace  the  following  curves  : 

(I)  y^=:  x4(l  -  x2).  (2)  y^  =  a-.'2(l  -  x).  (3)  x^  -  4  x^y  -  2 xy'^ -\- 4  y^  =  0. 
(4)  'iy'^  =  4xy  -  x^.      (5)  r  =  a  cos  4  d. 

*  A  recent  important  work  on  curves  is  Loria's  Special  Plane  Curves,  a 
German  translation  of  which  (xxi.  -f  744  pp.)  is  published  by  B.  G.  Teubner, 
Leipzig. 


CHAPTER    XIX. 

INFINITE   SERIES. 

EXPANSION  OF  FUNCTIONS  IN  INFINITE  SERIES.  INTEGRATION 
AND  DIFFERENTIATION  OF  INFINITE  SERIES.  EXPANSIONS 
OBTAINED  BY  INTEGRATION  AND  DIFFERENTIATION. 

N.B.  There  are  some  students  whose  time  is  limited  and  who  require  to 
obtain  as  speedily  as  may  be  a  working  knowledge  of  Taylor's  and  Maclaurin's 
expansions.  These  students  had  better  proceed  at  once  to  Arts.  175,  180, 
work  the  examples  in  Arts.  176  and  178,  and  then  take  up  Art.  174.    It 

is,  perhaps,  advisable  in  any  case  to  do  this  before  reading  this  chapter  and 
the  other  articles  in  Chapter  XX.  Those  who  are  studying  the  calculus  as  a 
"culture"  subject  should  become  acquainted  with  the  ideas  and  principles 
described,  or  referred  to,  in  Chapters  XIX.,  XX.  A  thorough  understanding 
of  these  ideas  and  principles  is  absolutely  essential  for  any  one  who  intends 
to  enter  upon  the  study  of  higher  mathematics. 

167.  Infinite  series :  definitions,  notation.  An  infinite  series 
consists  of  a  set  of  quantities,  infinite  in  number,  which  are  con- 
nected by  the  signs  of  addition  and  subtraction,  and  which  suc- 
ceed one  another  according  to  some  law.  A  few  infinite  series  of 
a  simple  kind  occur  in  elementary  arithmetic  and  algebra. 

For  instance,  the  geometrical  series 

the  geometrical  series 

1 -\- X  +  x^ -\- 1- x"-i  +  x«  +  a;«+i  +  ...,  (2) 

which  may  also  be  obtained  by  performing  the  division  indicated  in 
the  geometrical  series 


1  -X 


1  -  a;  +  ic2  +  ...  +  (-  l)nxn-i  +  ...,  (8) 

which  may  also  be  obtained  by  performing  the  division  indicated  in  ; 

the  geometrical  series  "^  ^ 

a  -\-  ar  -\-  ar"^  4-  •••  +  «r"-i  -f  ar"  +  ar^+^  +  ••• ;  (4) 

the  series                          l-f.J--)-A^ f-i_a.....  (5) 

IP     2p     Sp  nP 

300 


167,168.]  INFINITE  SERIES.  301 

The  successive  quantities  in  an  infinite  series,  beginning  with 
the  first  quantity,  are  usually  denoted  by 

Uq,  Ui,  U2y  •••,  w„_i,  ic„,  Un+i,  •••; 

or,  in  order  to  show  a  variable,  x  say,  by 

iio(x),  Ui(x),  ii.2(x),   ...,  w„_i(:c),  u^(x),  n^+i(x),  •-. 

Then  the  series  is 

Uq  -f-  U^  +  W2  H +  ««-l  H-  «n  +  ^«,.+l  H •  (6) 

The  value  of  the  series  is  often  denoted  by  s ;  and  the  symbol  s„ 
is  generally  used  to  denote  the  sum  or  value  of  the  series  obtained 
by  taking  the  first  n  terms  of  the  infinite  series ;  thus, 

Sn  =  Uo  -\-Ui-{-U2-\ h  ^in-l- 

The  valtie  of  the  infinite  series  (6)  is  the  limit  of  the  sum  of  the 
quantities  in  the  series;  i.e.  the  value  of  the  series  is  the  limit  of 
the  sum  of  n  terms  of  the  series  when  7i  increases  beyond  all 
bounds.*     This  is  expressed  in  mathematical  symbols 

s  =  lim„ix  Su.  (7) 

(This  limit  s  is  frequently,  but  not  quite  correctly,  called  "  the 
sum  of  the  series"  or  "the  sum  of  the  series  to  infinity^") 


Thus,  in 

(1), 

— M— i^-(-i^> 

and  hence 

s  =  \imn=ooSn  =  2; 

(7) 

in  (2), 

s„  =  1  +  X  +  x2  +  ...  +  a;"-i  =  ?^^^, 

x-1 

and  hence 

s  =  lim„i„  Sn  =  cc  when  x-^1  and  x  "^  —  1, 

(8) 

=  - — -  when   -  1  <  x<  1. 

(9) 

168.  Questions  concerning  infinite  series.  The  subject  of  infinite 
series  is  highly  important  in  mathematics.  Such  questions  as  the 
following  arise  and  require  to  be  answered : 

(a)  Under  what  conditions  may  infinite  series  be  employed  in 
mathematical  investigation  and  used  in  practical  work  ? 

*  Thus  s  is  not  the  sura  of  an  infinite  number  of  terms  of  the  series,  but  is 
the  limiting  value  of  that  sum. 


302  INFINITESIMAL   CALCULUS.  [Ch.  XIX. 

{h)  Under  what  conditions  may  an  infinite  series  be  used  to 
define  a  function  or  employed  to  represent  a  function? 

ThuSv  in  Art.  167,  result  (8)  shows  that  series  (2)  does  not  represent  the 

function  when  x  is  greater  than  1  or  less  than  —1  or  equal  to  1  or  —1. 

1  —  X 

This  is  obvious  on  a  glance  at  the  series ;  in  fact,  the  greater  the  number  of 
terms  of  (2)  that  are  taken,  the  greater  is  the  error  comjnitted  in  taking  the 
series  to  represent  the  function.  (For  instance,  put  x  =  2  ;  then  the  func- 
tion is  —  1  and  the  series  is  +  go.)     On  the  other  hand,  the  infinite  series  (2) 

does  represent  the  function    when  x  lies  between  —  1  and  +  1 ;  the 

\  —X 

greater  the  number  of  terms  that  are  taken,  the  more  nearly  will  the  sum  of 

these  terms  come  to  the  value  of  the  function.     The  limit  of  the  sum  of  these 

terms  when  the  number  of  them  is  infinite  is  the  function. 

(c)  May  two  infinite  series  be  added  like  two  finite  series  ?     In 

other  words,  if 

U  =  Ui^-\-  III  -f-  ^^2  +  •  •  • 

and  V  —  Vq  -\-V1-i-V2-] , 

is  u -\-v  =  no  +  VQ-{- n^-^Vi+  •"  (1) 

a  true  equation;  and  under  what  conditions  is  (1)  a  true  equation? 

(d)  May  two  infinite  series  be  multiplied  together  like  two 
finite  series  ?     In  other  words,  u  and  v  being  as  in  (c),  is 

a  true  equation;  and  under  what  conditions  is  (2)  a  true  equation  ? 

(e)  May  the  principles  of  Art.  31  and  Art.  104  A,  namely,  that 
the  derivative  and  the  integral  of  the  sum  of  a  Jinite  number  of 
terms  are  respectively  equal  to  the  sum  of  the  derivatives  and  the 
sum  of  the  integrals  of  these  terms  (to  a  constant),  be  extended 
to  infinite  series  ?     That  is,  Kq,  Ui,  Uo,  •••,  being  functions  of  x^  if 

Jsdx=  I    v^/Jx -\-  I    n^dx -\-  |    u^dx -\- -",  (3) 


are 


and  A  {s)  =  |-  («o)  +  f  (ti,)  +  f  {u.^  +  •  •  •,  (4) 

dx  dx  dx  dx 


ItjU.]  INFINITE   SERIES.  303 

true  equations;  and  what  are  the  conditions  which  must  be 
satisfied  in  order  that  these  equations  be  true  ?  Equations  (3)  and 
(4)  may  be  expressed : 

r     lim„-ao  s,,{x)  \(lx  =  lim„^3,    J     sXx)dx  L 

|[li.n„_^  «„(.•)]  =  lim_[|«„(.)]- 

The  above  questions  then  may  be  stated  thus :  Is  the  integral 
of  the  limit  of  the  sum  of  an  infinite  number  of  quantities  equal  to 
the  limit  of  the  sum  of  the  integrals  of  the  quantities ;  and  is  it 
likewise  in  the  case  of  the  differentials  ? 

For  instance,  given  that         =  1  +  x  +  ^^  +  a^"'  +  •••, 

I  —  X 

dx\l-xll        (l-x)2j 

and  is  f ' -^^  [i.e.  log  --1-]  =  x  +  f  +  ^'  +  -  ? 

Jo  1  _  XL  1  —  xj  2       8 

169.  Study  of  infinite  series.  Knowledge,  elementary  knowledge  at 
least,  of  the  theory  of  infinite  series,  and  practice  in  their  use  are  necessary  in 
applied  mathematics.  Infinite  series  frequently  present  themselves  in  the 
theory  and  applications  of  the  calculus,  and  accordingly  the  subject  should 
be  studied,  to  some  extent  at  least,  in  an  introductory  course  in  calculus. 
The  better  text-books  on  algebra,  for  instance,  among  others,  Chrystal's 
Algebra  (Vol.  II.,  Ed.  1889,  Chap.  XXVI.,  etc.),  Hall  and  Knight's  ^j^/ier 
Algebra  (Chap.  XXI.),  contain  discussions  on  infinite  series  and  examples  for 
practice.*  Osgood's  pamphlet,  Introduction  to  Infinite  Series  (71  pages, 
Harvard  University  Publications),  gives  a  simple,  elementary,  and  excellent 
account  of  infinite  series.  "This  pamphlet  is  designed  to  form  a  supplemen- 
tary chapter  on  Infinite  Series  to  accompany  the  text-book  used  in  the  course 
in  calculus."  Becent  text-books  on  the  calculus,  in  particular  those  of 
McMahon  and  Snyder,  Lamb,  and  Gibson,  contain  definitions  and  theorems 
on  infinite  series ;  they  will  especially  well  repay  consultation.  More 
elaborate  expositions  of  the  properties  of  infinite  series,  which  form  parts  of 
introductory  courses  in  modern  higher  analysis,  are  given  in  Harkness  and 
Morley,  Introduction  to  the   Theory  of  Analytic   Functions,    in  particular 

*  Also  see  Hobson,  A  Treatise  on  Plane  Trigonometry,  Chap.  XIV.,  and 
following  chapters. 


304  INFINITESIMAL    CALCULUS.  [Ch.  XIX. 

Chaps.  VIII. -XI.,  and  in  Whittaker,  Modern  Anali/sis,  in  particular  Chaps. 
II. -VIII.  These  discussions  can  be  read,  in  large  part,  by  one  who  possesses 
a  knowledge  of  merely  elementary  mathematics. 

A  statement  of  a  few  of  the  principal  definitions  and  theorems  which  are 
necessary  for  an  elementary  use  of  infinite  series  is  given  in  Arts.  170-173. 

170.   Definitions.      Algebraic   properties   of    infinite   series.      An 

infinite  series  has  been  defined  in  Art.   1()7.     If  (see  Art.  167) 

lim„^^  ,s„  is  a  definite  finite  quantity,  U  say,  tlie  series  is  called 

a  convergent  series,  and  is  said  to  converge  to  the  value  U.     If  s„ 

does    not    approach    a  definite  finite   value   when   n  approaches 

infinity,  the  series  is  called  a  divergent  series.     In  a  divergent 

series,  when  n  approaches  infinity,  s^  may  either  approach  infinity, 

or  remain  finite  but  approach  no  definite  value. 

Thus,  in  Art.  i:;7,  series  (1)  is  convergent;  series  (2)  is  convergent  for 

values  of  x  between  —  1  and  +  1,  for  then  s  =  — —  ;  series  (4)  is  convergent 

1—x 

when  r  lies  between  —  1  and  +  1,  for  then  s  =  — ^     .     Series  (5)  is  con- 

\  —  r 
vergent  for  j)  >  1,  and  divergent  for  jj  =  1  and  for  j9  <  1.      (Hall  and  Knight, 
Algebra^  p.  235.) 

[Note  1.      The  harmonic  series.     When  j?  =  1,  series  (5)  is 

2      3      4      5  w      n  +  1 

This  series  is  called  the  harmonic  series.'] 

The  series  1  +  2  +  3  + h  «  +  •••  is  divergent.     The  series  1  —  1  +  1  —  1  + 

•  ••  +  (—  1)*^"^  +  •••,  obtained  by  putting  x  =  1  in  series  (3),  is  divergent ;  for 
its  limit  is  0  or  1  according  as  n  is  even  or  odd.  (A  series  that  behaves  like 
this  is  said  to  oscillate.  Some  writers  do  not  include  oscillatory  series  among 
the  divergent  series.) 

In  general  only  convergent  series  are  regarded  as  of  service  in 
applied  mathematics.  (For  the  necessity  of  the  qualifying  phrase 
"  in  general,"  see  Note  2.)  A  series  may  be  employed  to  represent 
a  function,  or,  what  comes  to  the  same  thing,  a  function  may  be 
defined  by  a  series,  if  the  series  is  convergent.     Thus  series  (2), 

Art.  167,  may  be  used  to  represent  or  to  define ,  if  x  lies 

between  —  1  and  +  1.     [See  questions  (a)  and  (6),  Art.  168.*] 

*  Carl  Friedrich  Gauss  (1777-1855),  the  great  mathematician  and  astrono- 
mer of  Gottingen,  and  A ugus tin-Louis  Cauchy  (1789-1857),  professor  at  the 


170.]  INFIISITE  SERIES,  805 

Note  2.  Oil  divergent  series.  Those  who  apply  mathematics,  astroVio- 
mers  in  particular,  have  frequently  obtained  sufficiently  good  approximations 
to  true  results  by  means  of  divergent  series.  Such  series,  however,  "  cannot, 
except  in  special  cases,  and  under  special  precautions,  be  employed  in  mathe- 
matical reasoning"  (Chrystal,  Algebra,  Vol.  11. ,  p.  102).  At  the  present 
time  considerable  attention  is  being  paid  by  mathematicians  to  divergent 
series  and  to  investigations  of  the  fundamental  operations  of  algebra  and  the 
calculus  upon  them.  A  work  on  the  subject  has  recently  appeared,  viz. 
Leqons  sur  les  series  divergentes,  par  Emile  Borel  (Paris,  Gauthier-Villars, 
1901,  pp.  vi  +  182).  "It  is  safe  to  say  that  no  previous  book  upon  diver- 
gent series  has  ever  been  written."  Interesting  and  instructive  information 
concerning  divergent  series  will  be  found  in  reviews  on  this  book,  by  G.  B. 
Mathews  {Nature,  Nov.  7,  1901),  and  E.  B.  Van  Vleck  {Science,  March  28, 
1902). 

Absolutely  convergent  series.  A  series  the  absolute  values  (see 
Art.  8,  Note  1)  of  whose  terms  make  a  convergent  series  is  said 
to  be  absolutely  or  unconditionally  convergent;  other  convergent 
series  are  said  to  be  conditionally  convergent. 

Ex.  1.    Series  (1),  Art.  167,  is  an  absolutely  convergent  series. 

Ex.  2.    The  series  1  -  2  +  i  -  4  +  i («) 

may  be  written  (1  _  i)  +  (^  _  ^)  +  (|  -  |)+ ...,  i.e.  ^ +^1^  +  ^1^+ .... 

Series  (a)  may  also  be  written 


Thus  the  value  of  the  series  (a),  the  terms  being  taken  in  the  order  indi- 
cated, is  less  than  1  and  greater  than  \.  It  can  also  be  shown  that  this  series 
converges  to  a  definite  value.  On  the  other  hand  (see  Note  1,  and  the  state- 
ment just  preceding  Note  1),  the  series 

is  divergent.     Thus  series  (a)  is  a  conditionally  convergent  series. 

Theorems.  (1)  If  a  series  is  absolutely  convergent,  it  is  obvious 
that  any  series  formed  from  it  by  changing  the  signs  of  any  of 
the  terms  is  also  convergent. 

Polytechnic  School  at  Paris,  who  did  much  to  make  mathematics  more  rigor- 
ous than  it  had  been  during  its  rapid  development  in  tlie  eighteenth  century, 
may  be  regarded  as  the  founders  of  the  modern  theory  of  convergent  series. 
James  Gregory,  professor  of  mathematics  at  Edinburgh,  introduced  the  terms 
convergent  and  divergent  in  connection  with  infinite  series  in  1668. 


306  INFINITESIMAL   CALCULUS.  [Ch.  XIX. 

(2)  In  a  conditionally  convergent  series  it  is  possible  to  rearrange 
the  terms  so  that  the  new  series  will  converge  toward  an  arbitrary 
preassigned  value. 

(3)  In  an  absolutely  convergent  series  the  terms  can  be  rearranged 
at  pleasure  without  altering  the  value  of  the  series. 

(4)  If  (see  Art.  168)  u  and  v  are  any  two  convergent  series,  they 
can  be  added  term  by  term ;  that  is,  Equation  (1),  Art.  168,  is  true. 

(5)  If  u  and  v  are  any  two  absolutely  convergent  series,  they 
can  be  multiplied  together  like  sums  of  a  finite  number  of  quanti- 
ties ;  that  is.  Equation  (2),  Art.  168.,  is  true. 

For  proofs  and  examples  of  these  theorems  see  Osgood,  Intro- 
duction to  Inji)dte  Series,  Arts.  34,  35;  Chrystal,  Algebra,  Vol.  II., 
Chap.  XXVI.,  §§  12-14. 

In  a  convergent  series  as  n  increases,  s„  may  either:  (a)  con- 
tinually increase  toward  the  limiting  value  of  the  series ;  or 
(&)  decrease  toward  this  limit;  or  (c)  be  alternately  greater  than 
and  less  than  its  limit. 

Thus  iu  series  (1),  Art.  167,  Sn  continually  increases  toward  its  limit  (2); 

in  the  series  1— -+  — 1-  •••,  Sn  is  alternately  greater  than  and  less  than 

its  limit  |.  ^     ^"^     ^^ 

Remainder  after  n  terms.  The  symbol  Vn  or  Mn  is  often  used  to 
denote  the  series  (and  also  to  denote  the  value  of  the  series) 
formed  by  taking  the  terms  after  the  nt\\,  thus 

^"«  =  U,,  -f  U^^^  -f  Un+2  H • 

This  is  usually  called  the  remainder  after  n  terms.  Let  a  func- 
tion be  represented  by  a  convergent  series;  i.e.  let  the  value  of 
the  function  be  equivalent  to  the  value  of  this  convergent  series. 

Then  since  ^i     j?       x.-  ^^ 

the  lunction  =  lim„^  s„, 

it  follows  that  lim^^^  i\,  =  0. 

Interval  of  convergence.  In  general  a  convergent  series,  in  a 
variable,  x  say,  is  convergent  only  for  values  of  x  in  a  certain 
interval,  say  from  x  =  a  to  x  =  h.  The  series  is  then  said  to  con- 
verge within  the  interval  (a,  h),  and  this  interval  is  called  the 
iyiterval  of  convergence. 


171.]  INFINITE  SERIES.  307 

Thus  in  series  (2),  Art.  1(57,  the  interval  of  convergence  extends  from 
x  =  — ltoaj  =  +  l.  In  this  case,  as  in  many  others,  the  series  is  not  conver- 
gent for  the  values  of  x  (in  this  case  —  1  and  +  1)  at  the  extremes  of  the 
interval.  In  some  cases  series  are  convergent  for  the  values  of  the  variable  at 
the  extremes  of  the  interval  of  convergence  as  well  as  for  the  values  between  ; 
in  other  cases  a  series  may  be  convergent  for  the  value  of  the  variable  at  one 
extreme  of  the  interval  but  not  for  the  value  at  the  other. 

Power  series.    Series  of  the  type 

aQ  +  ciioc  -T  aox'^  +  •••  +  a^cc'*  —, 

in  which  the  terms  are  arranged  in  ascending  integral  powers  of  x 
and  the  coefficients  are  independent  of  Xj  are  called  power  series 
in  X.  A  power  series  may  converge  for  all  values  of  x,  but  in 
general  it  will  converge  for  some  values  of  x  and  diverge  for  others. 

Theorem.  In  the  latter  case  the  interval  of  convergence  ex- 
tends from  some  value  x  =  —  r  to  the  value  x  =  -{-  v]  i.e.  the  value 
ic  =  0  is  midway  between  the  values  of  x  at  the  extremes  of  the 

Divergent  Convergent  Divergent 

-r  o  +^' 

Fig.  101. 

interval  of  convergence.     Thus  in  the  power  series  (2),  Art.  167, 
the  interval  of  convergence  extends  from  —1  to  +1.     This  theo- 
rem may  be  graphically  represented,  or  illustrated,  by  Fig.  101. 
(For  proof  of  the  theorem  see  Osgood,  Lifinite  Series,  Art.  18.) 

171.  Tests  for  convergence.  Two  simple  tests  for  convergence 
will  now  be  shown.  For  nearly  all  the  infinite  series  occurring 
in  elementary  mathematics  these  tests  will  suffice  to  determine 
whether  a  series  is  convergent  or  divergent.  These  two  tests  are : 
(A)  the  comparison  test  and  (B)  the  test-ratio  test. 

A,  The  comparison  test.     Let  there  be  two  infinite  series, 

Wo+"l  +  '^2H h^n-l  +  ?*„+•••,  (1) 

and  Vq  -f  Vi  +  Vo  H 1-  ^'„-l  +  ^'«  H •  (2) 

If  series  (1)  is  convergent,  and  if  each  term  of  series  (2)  is  not 
greater  than  the  corresponding  term  of  series  (1)  {i.e.  if  v,^  ^  ?/„ 
for  each  value  of  ?i),  then  series  (2)  is  convergent.     If  series  (1) 


308  INFINITESIMAL   CALCULUS.  [Ch.  XIX. 

is  divergent,  and  if  each  term  of  series  (2)  is  greater  than  the 
corresponding  term  of  series  (1),  then  series  (2)  is  divergent. 
Two  series  which  are  very  useful  for  purposes  of  comparison  are : 

(a)    The  geometric  series 

a-{-  ar  -\-  wi^  -\-  ••♦, 
which  is  convergent  whein  |  ?•  |  <  1,  divergent  when  |  ?- 1  >  1. 

(6)   The  series  1 +;^  +  ^  4-:^  +  •••, 

which  is  convergent  when  p  >  1,  divergent  when  2)^1  (see  Art. 
170). 

Ex.  1.    The  series  i  +  i  +  ^^  +  ^Jg  +  ... 

is  convergent,  for  it  is  term  by  term  not  greater  tlian  the  geometric  con- 
vergent series  -.       ,        ,         , 

B,  The  test-ratio  test.     In  series  (6),  Art.  167,  the  ratio 


(3) 


is  comlnonly  called  the  test-ratio.  If  when  n  increases  beyond  all 
bounds  this  ratio  approaches  a  definite  limit  which  is  less  than  1, 
then  the  series  is  convergent.  For,  suppose  that  ratio  (3)  is  finite 
for  all  values  of  n,  and  suppose  that  after  a  certain  finite  number 
of  terms,  say  m  terms,  it  is  less  than  a  fixed  number  E  which  is 
less  than  1.     Now 

The  sum  of  the  first  m  terms  is  finite.     Since 

it  follows  that  the  series  beginning  with  u^  is  less  than  the 
geometric  series  /i   ,    n  .    r>2  .       \ 

and,  accordingly,  is  less  than 

1 


171.  J  INFINITE  SERIES.  309 

Hence  s<s^-^u^  ET^' 

and  thus  the  series  is  convergent. 

If  when  71  increases  beyond  all  bounds  the  test-ratio  approaches 
a  definite  limit  which  is  greater  than  1,  the  series  is  divergent. 

Ex.  2.    Prove  the  last  statement. 

If  the  limiting  value  of  the  test-ratio  is  4-  1  or  —  1,  further  special  investi- 
gation is  necessary  in  order  to  determine  whether  the  series  is  convergent  or 
divergent.* 

Thus  the  quality  of  the  series,  as  regards  its  convergency  or 
divergency,  depends  upon 

lim   •    ^w+i. 


EXAMPLES. 

3.  Find  whether  the  following  series  are  convergent  or  divergent : 

^  ^  2!3!4!o!  '    ^  ^   1p      2p     Sp     4p 

(5)   1+I+l  +  l  +  i-.... 
2p      oP      4p      5p 

4.  Examine  the  following  series  for  convergency  : 

(1)  1  -\-3x-\-  6 x^ +  7x^-^-9 x*-\-  •■•,  (2)  12  +  22a:  +  3-^a;2^.42a.3^.52a;4_,....^ 

(5)   1+5  +  ^  +  ^  +  ...+ 


2       5       10  n^  + 


*  A  series  in  which  the  absolute  value  of  the  test-ratio  tends  to  the  limit 
unity  as  n  increases,  will  be  absolutely  convergent  if,  for  all  values  of  n  after 
some  fixed  value, 


1    I  ft 

this  absolute  value  <  1 ^^^—^ 

—  11 


where  c  is  a  positive  quantity  independent  of  n.     (For  a  proof  of  this  general 
theorem,  see  Whittaker,  Modern  Analysis,  Art.  13.) 


310  INFINITESIMAL    CALCULUS.  [Ch.  XIX. 

172.  Integration  of  infinite  series.  This  article  and  the  next 
will  consider  briefly  the  questions  asked  in  Art.  168  (e).  Princi- 
ples (a)  and  (6)  stated  below  are  assumed  in  what  follows.  These 
principles  and  the  theorem  following  are  enunciated  and  proved 
in  Osgood,  Infinite  Series,  Arts.  37,  38,  39. 

(a)  A  sufficient -condition  that  the  series  of  continuous  functions 

Uq{x)  +  Ui(x)  +  n2(x)  +  .-. 

represent  a  continuous  function  is  given  in  the  following  theorem  : 

If  u^^(x)  +  u^(x)  + . . .,  a^x^fi, 

is  a  series  of  continuous  functions  convergent  throughout  the  interval 
(a,  /?),  then  the  function  f(x)  represented  by  this  series  will  he  continu- 
ous throughout  this  interval,  if  a  set  of  positive  numbers  Mq,  Mi,  M2, 
•  ",  independent  of  x,  can  be  found  such  that 

(1)  I  u,,(x)  \^M^,       a<x^^,       n  =  0,  1,  2,  ... ; 

(2)  Mq-\-M^  +  Mi  H is  a  convergent  series. 

(h)  On  applying  condition  {a),  for  a  series  to  represent  a 
continuous  function,  to  power  series  it  is  found  that : 

A  power  series  represents  a  continuous  function  within  its  interval 
of  convergence.  The  function  may,  however,  become  discontinu- 
ous on  the  boundary  of  the  interval. 

Integration  of  infinite  series  terra  by  terra.  Suppose  that  a  con- 
tinuous function  f(x)  is  represented  throughout  the  interval 
(a,  ^)  by  an  infinite  series  of  continuous  functions,  thus, 

f{x)  =  tto(x)  +  ui(x)  4-  Uo(x)  +  ...,  (1) 

this  series  accordingly  being  convergent  for  all  values  of  x  in 
the  interval  from  x  =  a  to  x  =  /3:  it  is  required  to  determine  the 
condition  that 

J'»/3  /'IS  /'/S  /»i3 

f(x)dx=  I     UQ(x)dx-\-  I    Ui(x)dx-{-  I    U2(x)dx-\-  ■"       (2) 

be  a  true  equation. 


172.]  INFINITE  SERIES.  311 

The  second  member  of  (2)  is  called  the  term  by  term  integral  of 
the  series. 

Note  1.  Professor  Osgood  in  an  article,  "  A  Geometrical  Method  for  the 
Treatment  of  Uniform  Convergence  and  Certain  Double  Limits"  {Bulletin 
of  the  Amer.  Math.  Soc^  2d  series,  Vol.  III.,  Nov.,  1896,  page  62),  gives  an 
instance  of  a  series  which  satisfies  the  conditions  imposed  on  (1),  but 
which  cannot  be  integrated  term  by  term. 

On  using  the  symbols  s„(x)  and  7'n{x)  as  defined  in  Arts.  1G7 
and  170,  Equation  (1)  can  be  written 

f(x)  =  s,Xx)  +  r„(.T). 
Accordingly,      j     f(x)dx=  I    s,Xx)dx-\-  I    r^ixjdx 

•Jo.  •Jo.  %J  a 

=  I     u^{x)dx-\-  I     Ux{x)dx-\ 

-h  j     n^_^{x)dx  4-  j    rXx)dx.  (3) 

Hence,  the  necessary  and  sufficient  condition  that  (2)  he  a  true 

equation  is  that  liin„^  j     i\{x)dx  =  0.  (4) 

It  can  be  shown  (see  Osgood,  Infinite  Series,  Art.  39)  that  this 
condition  is  satisfied  by  all  series  which  satisfy  the  conditions  of 
theorem  (a).     That  is  : 

Series  (1)  can  always  be  integrated  term  by  term,  i.e. 

Jf{x)  dx=  \     Wo(x)  dx  -f  I     Ui(x)  dx  -\-  \     n^ix)  dx  -{-  ••• 

will  be  a  true  equation,  if  a  set  of  positive  numbers,  Mq,  J/j,  Mo,  •••, 
independent  of  x,  can  be  found  such  that 

(1)  I  ^U^)  I  ^  ^K.       ^S^<  A       n  =  0,  1,  2,   ... ; 

(2)  Mq  -^  Ml  -\-  Mo  +  '"  is  a  convergent  series. 

This  test  is  "  sufficiently  general  for  most  of  the  cases  that 
arise  in  ordinary  practice." 


312  INFINITESIMAL    CALCULUS.  [Ch.  XIX. 

Note  2.  In  some  works  the  discussion  on  integration  of  infinite  series  is 
prefaced  by  an  explanation  of  wiiat  is  called  uniform  convergence,  and 
then  it  is  shown  that  uniformly  convergent  series  can  be  integrated  term  by 
term.  (E.g.  Gibson,  Calculus,  §§  151,  155,  which  contain  a  highly  com- 
mended general  treatment  of  integration  of  infinite  series,  and  which  should 
be  thoroughly  studied  by  all  who  are  interested  in  that  topic  ;  also  see  Lamb, 
Calculus,  Arts.  194-196,  where  the  integration  of  power  series  is  discussed.) 
It  is  shown  in  §  7  of  the  article  mentioned  in  Note  1  that  the  conditions 
imposed  on  the  series  in  theorem  (a)  constitute  a  sufficient  condition  for  the 
uniform  convergence  of  that  series. 

Application  to  power  series.  A  power  series  can  he  integrated 
term  by  term  throughout  any  interval  {a,  /?)  contained  in  the  interval 
of  convergence  and  not  reaching  out  to  the  extremities  of  this  interval. 

For  proof  of  this  theorem  and  for  other  applications  of  the 
preceding  test,  see  Art.  40  of  Osgood's  pamphlet. 

173.   The  differentiation  of  infinite  series,  term  by  term.     If  the 

function  f(x)   is  represented  by  the  series 

f(x)  =  u,(x)  +  u,{x)-{-"'  (1) 

throughout  the  interval  {a,  ^),  and,  if  moreover,  the  series  of  the 
derivatives  u,'(x) -\-u,'(x) -i- •.- 

satisfy  the  conditions  of  Theorem  (a),  Art.  172,  throughout  the 
interval  (a,  ^),  then  ivill  the  derivative  f'{x)  be  given  for  any  value 
of  X  in  the  interval  by  the  series  of  derivatives;  i.e.  then  will 

/'(a^)  =  <(a:)  +  <W  +  -  (2) 

he  a  true  equation  for  all  values  of  x  in  the  interval  (a,  (3). 
Let  the  series  of  derivatives  be  denoted  by  (l>{x)  ;  i.e.  let 

<f>{x)  =  u,'x-^u,'(x)-\-'-.  (.S) 

It  will  now  be  shown  that  <f>(x)  =f'(x). 

By  Theorem  (a),  Art.  172,  <f>(x)  is  continuous,  and  the  conditions, 
Art.  172,  for  the  term  by  term  integration  of  an  infinite  series  are 
satisfied.     Accordingly, 

J<f>(x)  dx=  i    Uq  (x)  dx  -\-  I    n^'  (x)  dx -] 
a                                «y  a                                    »./a 

=  [m„  (x)  -  ?<„  («)  +  K,  (a;)  -  Ml  (a)]  +  •  •  • 


173,  174.]  INFINITE  SERIES.  313 

On  differentiation,  <^(x)  =f'(x).     Hence,  Equation  (2)  is  true. 

By  the  aid  of  this  theorem  it  can  be  proved  that :  A  power  series 
can  be  differentiated  term  by  term  for  any  value  of  x  imthin,  but  not 
necessarily  for  a  value  at,  the  extremities  of  the  interval  of  conver- 
gence.    (For  proof  see  Osgood,  Infinite  Series,  p.  62.) 

Note  1.  For  instances  of  functions  defined  by  convergent  series  which 
cannot  be  differentiated  term  by  term,  see  Professor  Osgood's  article  men- 
tioned in  Note  1,  Art.  172. 

Note  2.  Articles  172,  173,  have  been  taken  in  large  part  from  Osgood's 
Introduction  to  Infinite  Series,  §  V.,  pp.  52-63.  Also  see  Lamb,  Calculus, 
Arts.  193-198  ;  Gibson,  Calculus,  §§  147-151,  155  ;  the  article  mentioned  in 
Note  1,  Art.  172,  §§  5-9. 

174.   Applications  of  the  integration  and  differentiation  of  series. 

A,  Expansions  obtained  by  integration  of  known  series.  Three 
important  examples  of  the  development  of  functions  into  infinite 
series  by  the  aid  of  integration,  will  now  be  given. 

EXAMPLES. 
Ex.  1.     For  -  l<.x<l 

1      =l_x-2  +  x4 .  (1) 


1  +  x^ 

i.e.  iRn-^x=x-^  +  ^-"'.  (2) 

o        o 

This  is  known  as  Gregory's  series.*  (For  complete  generality  the  term 
±mr,  (n=0,  1,  2,  •••),  should  be  in  the  second  member.)  Series  (1)  oscillates 
when  a;  =  1 ;  but  by  a  theorem  on  series  (see  Chrystal,  Algebra,  Vol.  II. , 
Chap.  XXVI.,  §  20)  series  (2)  is  convergent  and  represents  tan-^x  even 
when  X  =  1. 

Note  1.  Series  (2)  can  be  used  to  calculate  ir.  On  putting  x  =  1  in  (2), 
there  is  obtained  ,      ,      -, 

(a)   ^=1-1  +  1-14-.... 


*  Discovered  in  1670  by  James  Gregory  (1638-1675),  professor  of  matlie- 
matics  at  St.  Andrews  and  later  at  Edinburgh.  It  was  also  found  by  Leibnitz 
(1646-1716).  This  series  can  also  be  derived  independently  of  the  calculus 
(see  texts  on  Analytical  Trigonometry). 


314  INFINITESIMAL   CALCULUS.  [Ch.  XIX. 

This  is  a  very  slowly  convergent  scries.  More  rapidly  convergent  series 
for  calculating  t  are  the  following  : 

(6)    ^  =  4  taii-i-  -  tan-i—  :         (Machin's  Series  *) 
^  ^    4  5  239  ^  ^ 

(c)   -  =  tan-i-  +  tan-i-.  (Euler's  Series  t) 

4  Z  o 

Exercises.  Show  by  elementary  trigonometry  that  formulas  (ft)  and  (c) 
are  true.  Compute  the  value  of  ir  correctly  to  four  places  of  decimals : 
(1)  by  using  formula  (&)  and  Gregory's  series ;  (2)  by  using  formula  (c) 
and  Gregory's  series.  (The  correct  value  of  tt  to  ten  places  of  decimals  is 
3.1415926536.) 

Ex.  2.   For  -  l<x<l 


Vl-x2  2-4  2.4 

On  integrating  between  the  end  values  0  and  1,  as  in  Ex.  1,  there  results 

siii-icc  =  x  +  l.^'+l^.^  +  l^l^^^  +  .... 
2     3       2.4     5       2.4.G   7 

This  series  is  due  to  Newton,  and  was  used  by  him  in  computing  the  value 
of  TT.     When  x  —  I  this  series  gives 

TT      1  ,         1         ,         1.3         ,         1.3.5 


6      2      2  .  3  •  23      2  .  4  .  5  .  i;^      2  .  4  .  6  .  7  .  2" 

Exercise.  Using  the  last  result  calculate  tt  correctly  to  four  places  of 
decimals. 

Note  2,  For  historical  information  concerning  trigonometry  and  the 
computation  of  tt,  see  Murray,  Plane  Trigonometry,  Appendix,  Note  A,  and 
NoteC  (Art.  6);  Hobson,  article  "Trigonometry"  (Ency.  Brit.,  9th  edition); 
also  article  "  Squaring  the  Circle"  (Ency.  Brit.,  9th  edition). 

Ex.  3.    For  -  1  <  X  <  1 

:; =  1  —  X  -{■  X-  —  X^  -i-  •'-.  (1) 

1  +  X 


*  John  Machin,  died  1751,  was  professor  of  astronomy  at  Gresham  College, 
London. 

t  Leonhard  Euler,  1707-1783. 


174.]  INFINITE  SERIES.  315 

On  integrating  between  the  end  values  0  and  x,  as  in  Exs.  1, 2,  there  results 
log  (1  +  X)  =  ic  - 1  ic2  +  I  a?3  _  1  a?*  +  ....  (2) 

This  is  called  the  logarithmic  series.  *     (Here  the  base  is  e.) 

The  members  of  (2)  are  equal  for  values  of  x  as  near  1  as  one  pleases.  It 
is  also  easily  shown  that  they  are  finite  and  continuous  for  x  =  1.  Accord- 
ingly, formula  (2)  is  true  also  when  x=  \. 

On  putting  x  =  1  in  (2),  log  2  =  1  -  J  +  i  -  i  +  ...,  a  very  slowly  conver- 
gent series. 

On  putting  a;=-l  in  (2),  logO=- (1  +  1  +  1  +  1  +  .. .-)  =  -(».    (See  Art.  170.) 

Note  3.  Except  for  small  values  of  x  series  (2)  is  very  slowly  convergent. 
A  more  rapidly  convergent,  and  thus  more  useful,  series  for  the  computation 
of  logarithms  can  be  derived  from  (2),  as  follows.    On  putting  —  x  for  x  in  (2), 

log(l -x)  =-x- Ix2-ix3-ix* .  (3) 

.-.  log  1^  :=  2(x  +  1  x3  +  1  x5  +  ...).  (4) 

On  substituting  for  x  this  becomes 

*  2  w  +  1 

w^H  +  i^gf 1 1 1 n      ,5. 

^     m  L2m  +  1     3(2m  +  l)S^5(2m  +  l)5^     J'      ^^ 

Ifm  =  l,  log2  =  2fl  +  ^— +  -^  +  ...\=r.693. 

^  V3      3-33^5.35^      ) 

If  m  =  2,   log3  -  log2  =  2fl  +  ^  +  ^+  ...)  =  .406. 

/.  log3=:  1.099. 

Exercises.  (1)  Find  log  4  to  base  e,  by  putting  m  =  3  in  (5),  assuming 
the  value  of  log  3.  (2)  Find  the  logarithms  (to  base  e)  of  5,  6,  7,  8,  9,  10,  in 
a  similar  way.  (The  logarithms  of  4,  5,  6,  7,  8,  9,  10,  to  base  e,  to  three 
places  of  decimals,  are  respectively  1.386,  1.609,  1.792,  1.946,  2.079,  2.197, 
2.303.) 

B,   Expansions  obtained  by  the  differentiation  of  known  series. 

Following  is  an  example  of  the  development  of  functions  into 
infinite  series  by  means  of  differentiation,  the  conditions  of  Art. 
173  being  satisfied. 

*  Apparently  first  obtained  in  1668  by  Nicolaus  Mercator  of  Holstein. 


316  INFINITESIMAL    CALCULUS,  [Ch.  XIX. 

EXAMPLES, 


4.    When   -  1  <  x  <  1, 


1  -X 

On  differentiation, 

1 


^      -  1  +  a;  +  a;2  +  x3  +  ....  (1) 


1  +  2  a;  +  3  x^  4-  4  ajs  + 


(1-x; 
On  differentiation  and  division  by  (2), 

^ =  -(1.2  +  2.3x  +  3.4a;2+  ...). 

(1  -  x)^^     2  ^  ^ 

5.    Show  by  successive  differentiation  of  the  members  of  Ex.  4  (1)  that 

— i ..  (1  _  a;) -  =  1  +  mx  +  ^iill^iJzi)  a:2  +  m(m-1)(m-2) 

(1-x)-     ^  ^  1.2  1.2.3 

C  Approximate  integration  by  means  of  series.  Under  certain 
conditions  (Art.  172)  an  approximate  value  of  an  integral  may  be 
obtained  by  means  of  an  infinite  series.  Instances  have  been 
given  in  A  above. 


Thus,  in  Ex.  1,     r    "^aL       1_1+1      ...; 

*/o  1  4-  or  3      5 

dx 


^     o      C^      dx  1  ,        1         .        1-3 


EXAMPLES. 

6.  Find  an  approximate  value  of  the  length  of  the  ellipse  a;  =  a  sin  0, 
y  =  6  COS0.  [Here  0  is  the  complement  of  the  eccentric  angle  for  the  point 
(ic,  y).] 

Itrwill  be  found  (Art.  137)  that 

length  s  =:  4  «  I     Vl  —  e^  sin^  <p  d<t>.  (a) 

On  expanding  the  radical  by  the  binomial  theorem  and  taking  the  term  by 
term  integral  of  the  resulting  convergent  series  it  will  be  found  that 

'--[-Grf-(Hri-(i^rf--]-    <^) 

7.  Apply  result  (6)  of  Ex.  6  to  find  the  length  of  the  ellipse  whose  semi- 
axes  are  5  and  4.     (To  three  places  of  decimals.) 


174.]  INFINITE  SEEIES,  317 

8.    The  time  of  a  complete  oscillation  of  a  simple  pendulum  of  length  I, 
oscillating  through  an  angle  «(<  tt)  on  each  side  of  the  vertical,  is 


ii: 


Show  that  this  time 


"^  ,  in  which  k  =  sin  ^  a.  (c) 


-H['-(^?'-{^r^-{hur---} 


(d) 


Note  4.  Integrals  (c)  and  (a)  in  Exs.  8  and  6  are  known  respectively  as 
"elliptic  integrals  of  the  first  and  the  second  kind."  The  symbols  F{k,  0), 
E  (e,  0)  are  usually  employed  to  denote  these  integrals  (the  upper  end  value 
here  being  0).  Knowledge  of  these  integrals  was  specially  advanced  by 
Adrien  Marie  Legendre  (1752-1838).     See  Art.  122,  Note  4. 

9.    Show  that : 

..s  r^_ix_    1,1   1    1.3  1.1.3.5    1  , 
^0  ^/nr^         2   5    2.5   92.4.6   13       ' 


Vl  -  x^  ^5     2.59     2.4.6     13 

^^         =l_i.2_^1.2jL5.1__l.  2.5.8  ^  1. 


y/(mfiy  4    3      7     1-2    3-^      10     1-2.3     3^ 

dx       ^.^      1     1       1      1.4     1        1      1.4.7     1 
^/TZr^b  6*3      iri.2*32      16'l.2.3'33'^ 


CHAPTER   XX. 

TAYLOR'S   THEOREM. 
(See  N.B.  at  beginning  of  Chapter  XIX,) 

175.  Taylor's  theorem  is  one  of  the  most  important  theorems 
in  the  calculus.  It  has  a  wide  application,  and  several  important 
series,  for  example,  the  binomial  series  (see  Ex.  6,  Art.  176)  can 
be  derived  by  means  of  it.  Let  fi^x)  be  a  function  of  x  which  is 
continuous  throughout  the  interval  from  x  =  a  to  .v  =  h,  and  which 
also  has  all  its  derivatives  continuous  in  this  interval.  Now  let 
X  receive  an  increment  h.  Taylor\s  theorem  is  a  theorem  which 
gives  the  development  of  the  function  f{x  +  h)  in  a  power  series 
in  h.  The  power  series  itself  is  called  Taylor^s  series.  (See  Note 
2,  Art.  178.) 

N.B.    In  reading  this  chapter  it  is  better  to  take  up  Art.  180  first. 

176.  Derivation  of  Taylor's  theorem.  Analytic  proof  of  the 
theorem  of  mean  value.  Let  f{x)  and  its  first  derivative  be  con- 
tinuous in  the  interval   from  x  =  a  to  x  =  h.     Find  i^i  so  that 

/('>)-/(«)  =  (6 -a)Bi.  (1) 

Substitute  x  for  6,  and  put 

F(x)  =f(x)  -/(a)  -(x~  a)R,.  (2) 

Then  F(b)=0  by  (1);  also  F(a)  =  0  identically.  Hence,  by 
Eolle's  theorem  (Art.  63), 

F'(x,)  =  0, 

in  which  Xi  lies  between  a  and  h.     But  by  (2),  on  differentiation, 

F\x,)=f\x,)-R,. 
318 


175,176.]  TAYLOR'S   THEOREM.  319 

Accordingly,  E^  =f'{xi), 

as  already  shown  geometrically  in  Art.  64.     Hence,  from  (1),  on 
substituting  X  for  b,      ^^^^  ^^.^^^  ^  ^^  _  ^^^,^^^^^  ^3^ 

in  which  a^x^b,  and  a  <  Xi  <  a;. 
Result  (3)  may  be  written 

fix)  =f(a)  +  (x-  -  a)f[a  +  ^(o;  -  «)],      (4) 

in  which  0  <  ^  <  1.     Thus  the  theorem  of  mean  value  is  deduced 
analytically  from  Rolle's  theorem.     (See  Arts.  63,  64.) 

Taylor's  theorem.  Taylor's  theorem  can  be  derived  in  various 
ways.  The  method  adopted  in  this  article  is  merely  an  extension 
of  that  used  in  deriving  result  (4). 

Let /(if)  and  its  first  ?i-derivatives  be  continuous  in  the  interval 
from  x=a  to  x  =  b.     Find  i?„  so  that 


+  ^(t,l"i)";V'"""(")  )  =  (*  -  «)"'«»•  (5) 

Substitute  x  for  a,  and  let 

F(x)  =/(6)  -fix)  -  (6  -  x}f'(x)  -  i (6  -  xy-f"{x)  -  ... 

~  V-l)!'-^'"'"^"'^  "  ^*  ~  ''^"^'"  ^^^ 

Then  i^(a;)  is  continuous  in  the  interval  from  x  =  a  to  x  =  b, 
since  f{x)  and  its  first  7i-derivatives  are  continuous  there.  By  (5), 
F{a)  =  0 ;  also  F(b)  =  0  identically.     Hence,  by  Rolle's  theorem, 

in  which  a  <  iCj  <  6. 

But,  on  differentiation  and  reduction,  in  (6), 

F'ix,)  =  -^^=^^r(x,)+n(b-xy-^R„.  (7) 

in  which  0  <  ^  <  1. 


320  INFINITESIMAL    CALCULUS.  [Ch.  XX. 

On  substituting  this  value  of  R^  in  (5),  and  writing  x  for  a  and 
x-\-h  for  b,  there  is  obtained 

+  |^|/("Kaj  +  e^).  (9) 

This  is  Taylor's  theorem  with  the  remainder,  the  last  term  of  the 
second  member  being  denoted  as  the  remainder.  In  formula  (9) 
X  and  x-\-1i  must  both  be  in  the  interval  of  continuity;  in  any 
particular  application  of  this  formula,  x  has  a  fixed  value  and  h 
varies.  Theorem  [or  formula]  (9)  is  true  for  all  functions  which, 
with  their  first  ?i-derivatives,  are  continuous  in  the  assigned  inter- 
val of  continuity.  If  all  the  derivatives  of  f{x)  are  continuous  in 
the  interval,  and  if 

\im,J'^f^-\x-\-eh)  =  0, 

n ! 

then    /(a?  +  h)  =f(x)  +  hf'Coc)  +  ^f"(oo)  +  ^^/'"(jc)  +  ....     (10) 

For  (by  Art.  170)  the  infinite  series  in  the  second  member  converges 
to  the  value  of  f{x  +  h)  and,  accordingly,  represents  the  function 
f{x->th).  Formula  (10)  is  called  Taylor's  theorem,  and  the 
series  is  called  Taylor's  series.  In  (9)  and  (10)  h  may  be  positive 
or  negative,  so  long  as  x  and  x-\-h  are  in  the  interval  of  con- 
tinuity. "  Tlie  remainder,''^  the  last  term  in  (9),  represents  the 
limit  of  the  sum  of  all  the  terms  after  the  wth  term  of  the  infinite 
series  in  (10) ;  it  is  the  amount  of  the  error  that  is  made  when 
the  sum  of  the  first  «-terms  of  the  series  is  taken  as  the  value  of 
the  function. 

Note.  The  above  method  of  proving  the  theorem  of  mean  value  was  first 
given  by  Joseph  Alfred  Serret  (1819-1885),  professor  of  the  Sorbonne  in 
Paris,  in  his  Cours  de  calcul  differentiel  et  integral,  2e  ^d.,  t.  I.,  page  17  seq. 
The  above  proof  of  Taylor's  theorem  appears  in  Harnack's  Calculus  (Cath- 
cart's  translation,  Williams  and  Norgate),  pages  65,  66,  and  in  Gibson's 
Calculus,  pages  390-393.  The  proof  in  Echols's  Calc%ilus  (p.  82)  is  likewise 
based  on  the  theorem  of  mean  value. 

Taylor's  theorem  and  series  are  important  in  the  tlieory  of  functions  of 
a  complex  variable,  and  are  more  fully  investigated  iu  that  subject. 


176.]  TAYLOR'S  THEOREM.  321 


EXAMPLES. 
1.   Express  log  (x  +  h)  by  an  infinite  series  in  ascending  powers  of  h. 
Here  /(x  +  A)  =  log  {x^-h). 

.-./(x)  =loga:, 

X 

/'"(x)=4,  etc. 
x^ 

,.\og(x  +  h)=logx  +  'f-^  +  ^     J>^  +  .... 
X      2x2     3x^     4x* 

Here  x  must  not  be  0,  for  then  f(x)  =-  oo,  and  thus  is  discontinuous  for 
X  =  0.  The  series  is  evidently  more  rapidly  convergent  the  smaller  is  h  and 
the  larger  is  x. 

On  putting  x  =  1  and  h  =  1,  this  result  gives 

l0g2=:l-l+^-J+..., 

as  found  in  Ex.  3,  Art.  174. 

K  the  finite  series  in  (9)  is  used,  then 

log  (X  +  70  =  logx  +  ^  +  ^-^  +  ...  +  (-  l)"-i-T^-''-""T7T7^  0  <  ^<  1. 
X      2x2  ul(x  +  eh)» 

Here,  if  x  =  /i  =  1, 


log2  =  l -i  +  i-i  +  . ..+  (-!)«- 


w(i  +  ey 


On  interchanging  h  and  x  in  formula  (10),  if  that  can  be  done 
in  the  interval  of  continuity,  there  is  obtained  the  following 
form  of  Taylor's  theorem  : 

f(x  +  h)  =  f{h)  +  xf'ih)  + 1^  f'\h)  + 1^  f«'{h)  +  ...,       (11) 

a  form  which  is  often  useful.    Similarly  in  the  case  of  formula  (9). 
2.   Express  log  (x  +  K)  by  an  infinite  series  in  ascending  powers  of  x. 
Here/(x  +  /t)  =  log(x  +  /i).     •■  f{h)  =  \ogh,  f<{h)=\,  f<\h)^-l,  etc. 

...l0g(X+.)=l0g.  +  -^-^-f^3-.... 

lth  =  l,  log(l  +  x)=x-|  +  |-|4-..., 

as  already  obtained  in  Ex.  3,  Art.  174. 


322  INFINITESIMAL    CALCULUS.  [Ch.  XX. 

3.  Represent  sin  {x  +  h)  by  an  infinite  series  in  ascending  powers  in  h. 
Here  f{x  +  h)  =  sin  (x  +  h).    .-.  f{x)  =  sin  x,  f'(x)  =  cos  x,  /"(x)  =  -  sin  a:, 

etc. 

Hence,  on  using  formula  (10), 

7,2  7i3  Jji 

sin  (x  +  h)  =  sin  x  +  /*  cos  x sin  x cos  x  +  —  sin  x  +  •-. 

2!  3!  4! 

Let  X  =  J,  and  /i  =  ji^  of  a  radian  (i.e.  W  22".65). 
o 

Then 

f  -L  V  Z3  sin  '^  +  J-  cos  ^ 1 sin  ^ ^ cos  "^  +  . . .. 

,3      lOOy  3      100        3       (100)^2!        3       (100)33!        3 

This  is  a  rapidly  convergent  series. 

Now  sin  J  =  .86603,  cos  -  =  .50000.     On  making  the  computations,  it  will 

be  found  that,  to  Jive  places  of  decimals,  sin  60^'  34'  22". 65  =  .87099. 

Note.  The  last  exercise  is  an  example  of  one  of  the  most  useful  practical 
applications  of  Taylor's  theorem.  Namely,  if  a  value  of  a  function  is 
known  for  a  particular  value  of  the  variable,  then  the  value  of  the  function 
for  a  slightly  different  value  of  the  variable  can  he  computed  from  the  known 
value  by  Taylor^ s  formula.     (See  Art.  27,  Notes  1,3;  Art.  82,  Note  3.) 

4.  Expand  sin  (x  +  ^)  in  a  series  in  ascending  powers  of  x. 

In  this  case  form  (11)  is  to  be  used.  Here  /(x  + /i)  =  sin  (x  +  ^). 
.V  f{h)  z=  sin  h,  f'{h)  =  cos  h,  f"{h)  =  -  sin  h,  f"{h)  =  -  cos  h,  etc. 

.-.  sin  (x  +  h)  =  sin  h  +  x  cos  /i  —  —  sin  h  —  —  cos  h  +  •••. 

^  ^  2!  3! 

On  letting  ^  =  0,  the  following  important  series  is  obtained : 

sinx  =  x-  — +  — . 

3!      5! 

5.  Expand  cos  (x  +  h)  in  series,  (a)  in  ascending  powers  of  7i,  (6)  in 
ascending  powers  of  x.     From  the  latter  form  deduce  the  series 

qf.2  ^4 

cosx  =  l-— +  ^^ . 

2  !      4  ! 

6.  Expand  (x  +  /i)"*  by  Taylor's  formula  in  a  power  series  in  h,  and 
thus  obtain  the  Binomial  Expansion 

(X  +  h)^  =  x^  +  mx^-^h  +  ?^^^^^^  x^-2/12  +  .... 

1  ■  ^ 

(This  series  is  convergent  for  ^  <  1,  divergent  for  ^  >  1.     The  case  in  which 
h=±l  requires  special  investigation.) 


177.]  TAYLOR'S   THEOREM.  323 

7.  Given  that  f(x)  =ix^  -  Sx^+I x  +  5,  develop  f(x  +  2)  and  f(x -  3) 
by  Taylor's  expansion.  Then  find  /(x  +  2)  and  /{x  —  S)  by  the  usual 
algebraic  method,  and  thus  verify  the  results. 

8.  (1)  Assuming  sin  42°,  compute  sin  44°  and  sin  47°  by  Taylor's 
expansion.  (2)  Assuming  cos  32°,  compute  cos  34°  and  cos  37°  by  Taylor's 
expansion.     (3)  Do  further  exercises  like  (1)  and  (2). 

9.  Derive  \og<ix  + h)  =  \ogh +  1- ^  + ^- ]^  + -,  when  lx|<l; 

log(a:  +  /i)=.logx  +  ^--^  +  -^-...,  when  |x|>l. 
x     2x^     Sx^ 

10.    Show  that 

log  sin  (x-\-  a)=  log  sin  x  +  a  cot  x csc^  x  +  —    .— — h  •-••. 


177.  Another  form  of  Taylor's  theorem.  On  substituting  the 
value  of  B„  [Eq.  (8),  Art.  176]  in  (5)  and  writing  x  for  b,  there  is 
obtained 

^(a.-ar^:n)^-^^e(;r-a)].  (1) 

nl 

If  all  the  derivatives  of  f(x)  are  continuous  in  the  assigned 

interval,  and 

lim^  (^:zJ^/(n)[^  +  e(x  -  a)]  =  0, 
n  ! 

then  (Art.  170)  the  infinite  series/(a)  +  (a;-tt)/'(a)  +  i(a;-a)y"(a) 
+  •••  represents  the  f unction /(x)  *  ;  i.e. 

+  ^^^^/"\a)+....  (2) 

n  I 

Forms  (1)  and  (2)  for  Taylor's  theorem  and  series,  are  fre- 
quently useful.  The  last  term  in  the  finite  series  (1)  is  Lagrange^ s 
form  of  the  remainder  in  Taylor^ s  series.     (See  Note  4,  Art.  178.) 


Except  in  ^om§  rare  cases. 


324  INFINITESIMAL   CALCULUS,  [Ch.  XX. 


EXAMPLES. 

1.  Express  5  x^  4-  7  x  +  3  in  powers  of  x  —  2. 

Here              f(x)  =  5  x^  +  7  x  +  3,  .-.  /(2)  =  37, 

/'(x)  =  10x  +  7,  /'(2)=27, 

/"(x)  =  10,  /"(2)=10, 

f"'(x)=0,  /'"(2)=0. 

Now  by  (2),      /(x)  =/(2)  +  (x  -  2)/' (2)  +  (^  -  2)^/^2)  ^  .... 

.♦.  5  x2  +  7  X  +3  =  37  +  27(x  -  2)  +  5(x  -  2)2. 

2.  Express  4  x^  —  17  x^ -pll  x  +  2  in  powers  of  x  +  3,  in  powers  of 
X  —  5,  and  in  powers  of  x  —  4,  and  verify  the  results. 

3.  Express  by*  +  6  y^  —  11  y"^  -{-  ISy  —  20  in  powers  of  y  —  4  and  in 
powers  of  ?/  4-  4,  and  verify  the  results. 

Note.  Exs.  1-S  can  be  solved,  perhaps  more  rapidly,  by  Horner'' s  process. 
(See  text-books  on  algebra,  e.g.  Hall  and  Knight's  Algebra^  §  549,  4th  edition, 
1889.) 

4.  Develop  e*  in  powers  of  x  —  1. 

5.  Show  that   1=  l_ -l-(x- a)  +  i(x  -  a)2- l(x  -  a)34- -.,  when  x 

X     a     a^  a^  a* 

varies  from  x  =  0  to  x  =  2  a. 

6.  Show  that  log  x  =  (x  -  1)  -  ^(x  -  1)2  +  1  (x  -  1)3  -  ...  is  true  for 
values  of  x  between  0  and  2. 


178.  Maclaurin's  theorem  and  series.  This  is  a  theorem  for 
expanding  a  function  in  a  power  series  in  x.  As  will  be  seen 
presently,  it  is  really  a  special  case  of  Taylor's  theorem. 

Let  f{x)  and  its  lirst  n  derivatives  be  finite  for  x  =  0  and  be 
continuous  for  values  of  x  in  the  neighborhood  of  a;  =  0. 

In  form  (9),  Art.  176,  put  ic  =  0 ;  then 

fih)  =/(0)+7»/'(0)  + 1!/"(0)+ ...  +  ^-^/.-"(O)  +  ^/'\6h). 
On  writing  x  for  h,  this  becomes 

f{x)  =X0)  +x/'(0)+ 1^,/"(0)+ -  +  (-£^)-,/'"'"(0)  +  J/'-Xto).  (1) 


178.]  TAYLOR'S   THEOREM.  325 

lff(x)  and  all  its  derivatives  are  finite  for  x  =  0,  and  if 
lim,^^/"^%)  =  0,    then 


Ax)  =/(0)  +  acf'iO)  +  f^/"(0)  +  ...  +  ^/.nKO)  +  ....         (2) 
z  I  ft  I 

This  is  known  as  Maclanrin's  theorem,  and  the  series  is  called 
Maclanrin's  series.  The  last  term  in  (1)  is  called  the  remmnder  in 
Maclaiirin^s  series.  It  is  the  limit  of  the  sum  of  the  terms  of  the 
series  after  the  ;ith  term. 

EXAMPLES. 

1.  Show  that  formula  (2)  comes  from  form  (11),  Art.  176,  on  putting 
^  =  0  ;  show  that  this  has  practically  been  done  in  the  derivation  above. 
Show  that  formula  (2)  comes  from  form  (2),  Art.  177,  on  putting  a  =  0. 

2.  Develop  sin  x  in  a  power  series  in  x. 

Here  f(x)  =  sin  x.  .'.  /(O)  =  0, 

:.f>ix)=.cosx,  /'(0)  =  1, 

/"(x)=-sinx,  /"(0)  =  0, 

/"'(x)=-cosx,  /'"(0)=-l, 

/i^(a:)  =  sinx,  /iv(0)  =  0, 

etc.  etc. 

(Compare  Ex.  2  above  and  Ex.  4,  Art.  176.) 

On  applying  the  method  of  Art.  171  it  will  be  found  that  the  interval  of 
convergence  is  from  —  co  to  +  oo. 

3.  Calculate  sin  (^  radian),  i.e.  sin  5°  43'  46".5. 

By  A,  sin  (.1  radian)  =  .1  -  ^^  +  ^^ =  .09983. 

4.  Calculate  sin  (.5'')  and  sin  (.2'')  to  5  places  of  decimals.  (For  Tesults, 
see  Trigonometric  Tables.) 

5.  Show  that  cos  «  =  1  -—+  —  -  —  +••• ,  (B) 

2  !      4  !     6  ! 

and  show  that  the  interval  of  convergence  is  from  —  go  to  +  oo. 

6.  To  4  places  of  decimals  calculate  the  following:  sin(.3''),  cos (.2)'', 
sin  (.4''),  cos  (.4').     (See  values  in  Trigonometric  Tables.) 


326  INFINITESIMAL   CALCULUS.  [Cii.  XX. 

7.  Show  that  e*  =  1  +  X  +  f-  +  ^  +  .-,  (C) 
and  show  that  this  series  is  convergent  for  every  finite  value  of  x. 

8.  Substitute  1  for  x  in  C,  and  thus  deduce  2.71828  as  an  approximate 
vahie  of  e. 

9.  Assuming  A  and  B  deduce  that  the  sine  of  the  angle  of  magnitude  zero, 
is  zero,  and  that  the  cosine  of  this  angle  is  unity. 

Note  1.  Expansions  A  and  B  were  first  given  by  Newton  in  1669.  He 
also  first  established  series  C.  These  expansions  can  also  be  obtained  by  the 
ordinary  methods  of  algebra,  without  the  aid  of  the  calculus.  For  this 
derivation  see  Chrystal,  Algebra,  Part  II.,  Chap.  XXIX.,  §  14,  Chap. 
XXVIII.,  §  5,  and  the  texts  of  Colenso,  Hobson,  Locke,  Loney,  and  others, 
on  what  is  frequently  termed  Analytical  Trigonometry,  or  Higher  Trigo- 
nometry. [This  subject  is  rather  to  be  regarded  as  a  part  of  algebra 
(Chrystal,  Algebra,  Part  II.,  p.  vii).]  Also  see  article  "Trigonometry" 
{Ency.  Brit.,  9th  ed.). 

10.  Develop  the  following  functions  in  ascending  powers  in  x  :  (1)  sec  x  ; 
(2)  log  sec  x  ;  (3)  log  (1  +  x).     Compare  the  latter  with  Ex.  3,  Art.  174. 

11.  Show  that  tan  X  =  X  -I-  \x^  -\-  ^^  x^  +  g^  x"^  +  .... 
By  this  series  compute  tan  (.5''),  tan  15°,  tan  25°. 

12.  Find:   (1)    fe^cosxffe;    (2)    C^dx;   (3)    f%-^\?x. 

J  Ja    X  Jo 

Note  1  a.  The  integral  in  Ex.  12  (3)  is  important  in  the  theory  of  probabili- 
ties. If  the  end-value  x  is  oo,  the  value  of  the  integral  is  ^Vw.  (Williamson, 
Integral  Calculus,  Ex.  4,  Art.  116.) 

13.  Assuming  the  series  for  sin  x,  prove  Huyhen's  rule  for  calculating 
approximately  the  length  of  a  circular  arc,  viz. :  From  eight  times  the  chord 
of  half  the  arc  subtract  the  chord  of  the  whole  arc,  and  divide  the  result  by 
three. 

14.  State  Maclaurin's  theorem,  and  from  the  expansion  for  tanx  find 
the  value  of  tan  x  to  three  places  of  decimals  when  x  =  10°. 

15.  Show  that  cos«  x  =  1  -  -^  x2  +  '^^^  ^  ~  ^)  x* • 

2!  4! 

Note  2.  Historical.  Taylor''s  theorem,  or  formula,  was  discovered  by 
Dr.  Brook  Taylor  (1685-1731),  an  English  jurist,  and  published  in  his  Metho- 
dus  Incrementorum  in  1715.  It  was  given  as  a  corollary  from  a  theorem  in 
Finite  Differences,  and  appeared  without  qualifications,  there  being  no  refer- 
ence to  a  remainder.  The  formula  remained  almost  unnoticed  until  Lagrnnue 
(1736-1813)  discovered  its  great  value,  investigated  it,  and  found  for  the 


179.]  TAYLOR'S   THEOREM.  327 

remainder  the  expression  called  by  his  name.  His  investigation  was  pub- 
lished in  the  3Iemoires  de  V Academic  de  Sciences  a  Berlin  in  1772.  "Since 
then  it  has  been  regarded  as  the  most  important  formula  in  the  calculus." 

Maclaurin'^s  formula  was  named  after  Colin  Maclaurin  (1698-1746),  pro- 
fessor of  mathematics  at  Aberdeen  1718  ?-1725,  and  at  Edinburgh,  1725-174-5, 
who  published  it  in  his  Treatise  on  Fluxions  in  1742.  It  should  rather  be 
called  Stirling''s  theorem,  after  James  Stirling  (1690-1772),  who  first  an- 
nounced it  in  1717  and  published  it  in  his  3Iethodus  Differentialis  in  17o0. 
Maclaurin  recognized  it  as  a  special  case  of  Taylor's  theorem,  and  stated 
that  it  was  known  to  Stirling ;  Stirling  also  credits  it  to  Taylor. 

Note  3.  Taylor's  and  Maclaurin's  theorems  are  virtually  identical.  It 
has  been  shown  in  Art.  178  that  Maclaurin's  formula  can  be  deduced  from 
Taylor's.  On  the  other  hand,  Taylor's  formula  can  be  deduced  from  Mac- 
laurin's ;  e.g.  see  Lamb's  Calculus^  page  567,  and  Edwards's  Treatise  on 
Differential  Calculus^  page  81. 

Note  4.  Forms  of  the  remainder  for  Taylor's  series  (2),  Art.  (177). 
Lagrange's  form  of  the  remainder  has  already  been  noticed  in  Art.  177. 
Another  form,  viz.  • 

^x-aY(i      ^)'--^^(.)[-^  ^  Q^^_  ^)-|^  0<^<1, 

was  found  by  Cauchy  (1789-1857),  and  first  published  in  his  Leqons  sur  le 
Calcitl  infinitesimal  in  1826.  A  more  general  form  of  the  remainder  is  the 
Schlomilch- Roche  form,  devised  subsequently,  viz. 

(:,      ^).(1      ,)n-.^^^^^^  +  ^(o:  -  a)],  0  <  ^  <  1. 
(n  -\)\  p 

This  includes  the  forms  of  Lagrange  and  Cauchy  ;  for  these  forms  are  ob- 
tained on  substituting  n  and  1  respectively  for  p.  (The  0's  in  these  forms 
are  not  the  same,  but  are  alike  in  being  numbers  between  0  and  1.)  In  par- 
ticular expansions  some  one  of  these  forms  may  be  better  than  the  others  for 
investigating  the  series  after  the  first  n  terms. 

Note  5.  Extension  of  Taylor's  theorem  to  functions  of  two  or  more 
variables.  For  discussions  on  this  topic  see  McMahon  and  Snyder's  Calcu- 
lus^ Art.  103  ;  Lamb's  Calculus,  Art.  211  ;  Gibson's  Calculus,  §  157. 

Note  6.  References  for  collateral  reading  on  Taylor''s  theorem. 
Lamb,  Calculus,  Chap.  XIV. ;  McMahon  and  Snyder,  Diff.  Cal.,  Chap.  IV. ; 
Gibson,  Calculus,  Chaps.  XVIIL,  XIX.  ;  Echols,  Calculus,  Chap.  VI. 

179.  Relations  between  trigonometric  (or  circular)  functions  and  expo- 
nential functions.  The  following  important  relations,  which  are  extremely 
useful  and  frequently  applied,  can  be  deduced  from  the  expansions  for  sinx, 
cos  X,  and  e^  in  Art.  178. 


328  INFINITESIMAL   CALCULUS.  [Ch.  XX. 

The  substitution  of  ix  for  x  in  C  gives 

e**^  1-^  +  ^ ^i(x-^-\-^ ^  =  COS  a?  +  i  sin  ic.     (1) 

V        3  !      5  !  / 


2  !      4  ! 
The  substitution  of  —  ix  for  x  in  C  gives 


2  !      4  !  V        3  !      5  ! 

From  (1)  and  (2),  on  addition  and  subtraction. 


■  J  =  COS  a?  -  i  sin  x,    (2) 


cos£c  =  ^^2^ (3),  sina?=^-^| (4) 

On  putting  tt  for  x  in  (1),  there  is  obtained  the  striking  relation 

e**^  =  -  1.  (See  Art.  38,  Note  on  e.) 

Note  1.  The  remarkable  relations  (l)-(4),  by  which  the  sine  and  cosine 
of  an  angle  can  be  expressed  in  terms  of  certain  exponential  functions  of  the 
angle  (measured  in  radians),  and  conversely,  were  first  given  by  Euler 
(1707-1783).  (In  connection  with  the  expansions  in  Arts.  178,  179,  see  the 
historical  sketch  in  Murray's  Plane  Trigonometry,  Appendix,  Note  A  ;  in 
particular  pp.  168,  169. ) 

Note  2.  Results  (l)-(4)  can  also  be  deduced  by  the  methods  of  ordinary 
algebra;  see  Note  1,  Art.  178,  the  references  therein,  and  Chrystal's  Algebra, 
Part  II.,  Chap.  XXIX.,  §  23. 

EXAMPLES. 

1.  From  (3)  and  (4)  deduce  that  cos^  x  +  sin^x  =  1.     . 

2.  Show  that  tan  x  = 

3.  Express  cot  x,  sec  x,  cosec  x,  in  terms  of  exponential  functions  of  x. 

Note  3.  Since,  by  (1),  e'*  =  cos  0  +  i sin  0,  and  e'***  =  cos  n<p-\-  isin  n  (p, 
and  since  (e»*)«  =  e''*^  it  is  evident  that 

(cos  4>  4-  i  sin  4>)  **  =  cos  n^  +  i  sin  n^, 

for  all  values  of  n,  positive  or  negative,  integral  or  fractional. 

This  very  important  theorem  is  called  De  Moivre''s  theorem,  after  its  dis- 
coverer Abraham  de  Moivre  (1667-1754),  a  French  mathematician  who 
settled  in  England.  It  first  appeared  in  his  Miscellanea  Analijtica  (London, 
1730),  a  work  in  which  "he  created  'imaginary  trigonometry.'"  [On  De 
Moivre' s  theorem,  and  results  (l)-(4),  see  Murray,  Plane  Trigonometry, 
Art.  98,  and  Appendix,  Note  D  ;  and  other  text-books  on  Trigonometry.] 

N.B.    The  article  on  Hyperbolic  Functions,  Appendix,  Note  A,  may  be 

conveniently  read  at  this  time. 


180.]  TAYLOR'S   THEOREM,  329 

180.   Another  method  of  deriving  Taylor's  and  Maclaurin's  series. 

Following  is  a  method  which  is  more  generally  employed  than 
that  in  Arts.  176  and  178  for  finding  the  forms  of  the  series  of 
Taylor  and  Maclaurin. 

A,  Maclaurin's  series.  Let  f{x)  and  its  derivatives  be  con- 
tinuous in  the  neighbourhood  of  a;  =  0,  say  from  x  =  —  a  to  x  =  a. 
Suppose  that  f{x)  can  be  expressed  in  a  power  series  in  x  conver- 
gent in  the  interval  —  a  to  +  a.  That  is,  assume  that  (for 
—  a<Cx<d)  there  can  be  an  identically  true  equation  of   the 

fo^m       fix)  =  A  4-  A,x  -f-  A^-^  +  A^x^  -f  ...  -+  A^x-  +  .••.  (1) 

The  coefficients  A  A^,  A2,  ••,  A^,  •.•,  will  now  be  found.  It 
has  been  seen  in  Art.  173  that  if  Equation  (1)  is  identically  true, 
then  the  equation  obtained  by  differentiating  both  members  of  (1), 

viz.  ff^^)  =  A  +  2  A^  +  3  A,x'  4-  •  •  •  +  nA,,x^^-'  +  .  •  -, 

also  is  identically  true  for  values  of  x  in  some  interval  that 
includes  zero.  For  the  same  reason  the  following  equations, 
obtained  by  successive  differentiation,  are  also  identical  in  inter- 
vals that  include  zero,  viz. : 

/"(x)  =  2  A  +  2  .  3  Aa^  H-  ...  +  n(n  -  l)Aa^"'  +  -, 
f"(x)  =  2>S-A,-i-''-+n(n-l){n-2)A^x--'-\--, 

/(«)(a;)  =  ?i.?i-l  .?i-2 2.1  A+--, 


On  putting  x  =  0  in  each  of  these  identities  it  is  found  that 

A=/(0),  A,=r(0),  A,  =  ^,  ^3  =  -^,  -,  A  =  -^^,  -. 

Hence,  on  substitution  in  (1), 
/(x)=/(0)+  x/'(0)  +  |:/"(0)  +  g/"'(0)+  ...  +5!/x..(0)+  ...,     (2) 

which  is  Maclaurin's  series  (Art.  178). 

B,  Taylor's  series.  Let  f(x)  and  its  derivatives  be  continuous 
in  the  neighbourhood  of  x  =  a,  say  from  x  =  a  —  h  to  x  =  a-\-  h. 
Suppose  that  f(x)  can  be  expressed  in  a  power  series  in  cc  —  a 


330  INFINITESIMAL   CALCULUS.  [Ch.  XX. 

which  is  convergent  in  the  neighbourhood  oi  x  =  a.  In  other 
words,  suppose  that  there  is  an  identically  true  equation  of  the 
form 

fix)  =  A,  -^A,(x-a)  +  A,  (x  -  af  +  ^3  (^  -  a)^  +  •  •  • 

+  A(«^-«r  +  ---  (3) 

Then,  as  in  case  A,  the  following  equations,  which  are  obtained 
by  successive  differentiation,  also  are  identically  true  for  values 
of  X  near  x  =  a,  viz. : 

f'{x)=A,-\-2A,(x-a)-{-SA.,(x-ay-+"'-\-nA,(x-ay-^  +  "', 

f"(x)  =  2  A,  +  2  ■  3  Az(x-a) -^"- -^n- n-1  •  A^{x-  a)'^-2+ ..., 

f"{x)  =2-3-  A^-[-'"^n- n -1-71-2  ■  A^{x-  ay-'^  +  ••., 

f'{x)  =71-71-1  '71-2  -  ...2.1.  A+--, 

On  putting  .t  =  a  in  each  of  these  identities  it  is  found  that 


71 


Hence,  on  substitution  in  (3),        \ 
fix)  =f{a)  +{x-  a)f(a)  +  (^^f"(a)  +  •  •• 

+  ^^^^^/'"'(a)  +  -,  (4) 

711 

which  is  series  (2),  Art.  177. 

If  in  (4)  X  is  changed  into  x-^a,  then 

f(x  +  a)  =f(a)  +  xf'(a)  +  ^f"(a)  +  •  •  •  +  ^  f^^Ka)  +  • . .,  (5) 

which  is  series  (11),  Art.  176,  with  a  written  for  h.     On  inter- 
changing a  and  x  in  (5),  form  (10),  Art.  176,  is  obtained. 

Note.  On  the  proof  of  Taylor's  theorem.  The  above  merely  shows  the 
derivation  of  the  fonn  of  Taylor's  series.  It  is  still  necessary  to  examine  into 
the  convergency  or  divergency  of  the  series  and  to  determine  the  remainder 


181.]  TAYLOR'S   THEOREM.  331 

after  any  number  of  terms.  The  investigation  of  tlie  validity  of  the  series  is 
a  very  important  matter  in  the  calculus.  For  this  investigation  see,  among 
other  works,  Todhunter,  Diff.  Cal.,  Chap.  VI.  ;  Williamson,  Dif.  Cal., 
Arts.  73-77  ;  Edwards,  Treatise  on  Diff.  Cal.,  Arts.  130-142  ;  McMahon 
and  Snyder,  Diff.  Cal.,  Chap.  IV.  ;  Lamb,  Calculus,  Arts.  203,  204;  article, 
"Infinitesimal  Calculus"  (Ency.  Brit.,  9th  ed.,  §§  46-52). 

181.  Application  of  Taylor's  theorem  to  the  determination  of  con- 
ditions for  maxima  and  minima.  This  article  is  supplementary  to 
Art.  76.  Let  f(x)  be  a  function  of  x  such  that  f{a-[-li)  and  f(a  —  h) 
can  be  developed  in  Taylor's  series;  and  let  it  be  required  to 
determine  whether /(a)  is  a  maximum  or  minimum  value  of  f{x). 
On  developing  /(a  —  h)  and  f{a  -\-  h)  by  formula  (9),  Art.  176, 

f(a  -  h)  =f(a)  -  hf'(a)  +  |^/"(a)  - 1^ /'"(«)  +  - 


nl 


f{a  +  h)  =/(a)  -f-;i/'(a)  -h|^/"(a)  +|^/'"(a)  +  - 

+  A!/(«)(a  +  ^^,),  (2) 


in  which  6^  and  $2  lie  between  0  and  1. 

Suppose  that  the  first  n  —  1  derivatives  of  f{x)  are  zero  when 
x=a,  and  that  the  nth  derivative  does  not  vanish  for  x=a.   Then 

/(a-A)-/(a)  =  t-^/(»)(a_e,70,  (3) 

n ! 

/(a  -h  h)  -f{a)  =  ^ /">(a  +  O.Ji).  (4) 

It  follows  from  the  hypothesis  concerning /(a;)  that  the  signs  of 
/(">(a  —  eji)  and/("X«  +  Ooh),  for  infinitesimal  values  of  h,  are  the 
same  as  the  sign  of  /^"^(a).  From  (3),  (4),  and  the  definitions  of 
maxima  and  minima,  it  is  obvious  that : 

(a)  If  n  is  odd,  the  first  members  of  (3)  and  (4)  have  opposite 
signs,  and  consequently,  f(a)  is  neither  a  maximum  nor  a  minimum 
value  off(x)  ;  (6)  Ifn  is  even  and  p'\a)  is  positive,  the  first  mem- 
bers of  (3)  and  (4)  are  both  positive,  and  consequently,  /(a)  is  a 


332  INFINITESIMAL   CALCULUS.  [Ch.  XX. 

miniimim  vahie  of  f{x) ;  (c)  Ifn  is  even  and  p''\a)  is  negative,  the 
first  members  of  (3)  and  (4)  are  both  negative,  and  consequently, 
/(a)  is  a  maximum  value  of  f{x).  The  condition  for  maxima  and 
minima  that  was  deduced  in  Art.  76,  (c),  is  a  special  case  of  this, 
viz.  the  case  in  which  n  =  2. 

182.  Application  of  Taylor's  theorem  to  the  deduction  of  a  theorem 
on  contact  of  curves.  This  article  is  supplementary  to  Art.  14o. 
(See  Art.  143,  Note  4.) 

Theorem.  If  two  curves  have  contact  of  an  even  order,  they  cross 
each  other  at  the  jooint  of  contact;  if  tivo  curves  have  contact  of  an 
odd  order,  they  do  not  cross  each  other  at  the  point  of  contact. 

Let  the  two  curves   y  =  cf)(x)  and  y  =  if/(x)  (1) 

have  contact  of  the  nth.  order  at  a?  =  a.     Then 

<f>(a)  =  ^(a),  <f>'(a)  =  ^'(a),  <f>"(a)  =  ^"(a),  ...,  4>^^%a)  =  ^^^%a).    (2) 

Now  compare  the  ordinates  of  these  curves  at  x  =  a  —  h,  i.e.  com- 
pare cf>(a  —  h)  and  if/(a  —  h);  also  compare  the  ordinates  at  x  =  a  +  h, 
i.e.  compare  cf){a  +  h)  and  i/^(a  -f  h).  Let  it  be  further  premised 
that  cfi(a  ±  h)  and  i/'(a  ±  h)  can  be  expanded  in  Taylor's  series.  On 
using  Taylor's  theorem  (form  9,  Art.  176),  and  remembering 
hypothesis  (2),  it  will  be  found  that 

^{a -  70  -  ^(a  -  h)  =  t|^  [<^'"+" (a  -  0,K)  -  f  "«>(a  -  BX)\,  (3) 

^a  +  70  -  ^(a  +  A)  =  j^-^  [<^<"«>(a  -  6Ji)  -  ^"•+"(«  -  «.''•)].   (*) 

in  which  the  four  ^'s  all  lie  between  0  and  1. 

Let  h  approach  zero;  then,  by  the  premise  above,  the  signs 
Of  the  expressions  in  brackets  are  the  same  as  the  signs  of 
[<^("+^^(a)  -  i/^('*+i)(a)].  Hence,  ifn  is  odd,  the  first  members  of  (3) 
and  (4)  have  the  same  sign,  and,  accordingly,  the  curves  do  not 
cross;  if  n  is  even,  these  first  members  have  opposite  signs,  and, 
accordingly,  the  curves  do  cross. 

Ex.    Accompany  the  proof  of  this  theorem  with  illustrative  figures. 


182, 183.]  Taylor's  theorem.  338 

183.   Applications  of  Taylor's  theorem  in  elementary  algebra.    Let 

f(x)  be  a  rational  integral  function  of  x,  of  the  nth  degree  say. 
Then  p''^'^\x)  and  the  following  derivatives  are  all  zero.  Hence, 
Taylor's  series  for  f(x  +  h)  in  ascending  powers  of  either  h  or  x 
[see  forms  (10)  and  (11),  Art.  176]  is  finite.     That  is, 

/(*  +  /0=/(x)  +  ft/'(a.)  +  g/"(^)+-+^/<">(a;),  (1) 

f(x  +  h)=f(l,)  +  xfQi)  +  ^/'(h)+...  +  J/<"'('0-  (2) 

A  rational  integral  function  f{x)  of  the  ?ith  degree  can  also  be 
e:^pressed  in  a  finite  series  in  ascending  powers  of  x  —  a  [see 
form  (2),  Art.  177].     That  is, 

/(x)=/(a)  +  (a;-a)/'(a)  +  (-^^V"(«)+  -  +^^^>")(a).  (3) 

Exercise.     See  Ex.  7,  Art.  176,  and  Exs.  1,  2,  3,  Art.  177. 

Note  1.  Let /(x)  be  as  specified  above.  In  general  the  calculation  of 
f{x  +  h)  and  the  expression  of  f{x)  in  terms  of  x  —  a,  can  be  more  speedily- 
effected  by  Horner'' a  process*  This  process  is  shown  in  various  texts  on 
algebra;  e.g.  Hall  and  Knight's  Algebra  (4th  edition),  Arts.  549,  572. 

Note  2.    For  an  applicatiou  of  Taylor's  theorem  to  interpolation, 

see  McMahon  and  Snyder,  Calculus,  Note,  pp.  325,  326. 

Note  3.  In  expansion  (10),  Art.  176,  if  h  is  a  differential  dx  of  x,  then 
A,  Ti^,  h^^  ...,  are  respectively  differentials  of  x  of  the  first,  second,  third,  •••, 
orders;  and  hf(x),  h^f"(x),  Ay"(x),  •••,  are  respectively  differentials  of 
/(.f)  of  the  first,  second,  third,  •••,  orders.  If  h  (or  dx)  is  an  infinitesimal, 
these  differentials  are  also  infinitesimals  of  the  respective  orders  mentioned. 


*  William  George  Horner  (1786-1837),  an  English  mathematician,  who 
discovered  a  very  important  method  of  finding  approximate  solutions  of 
numerical  equations  of  any  degree. 


CHAPTER   XXI. 

DIFFERENTIAL   EQUATIONS. 

N.B.  The  references  made  in  this  chapter  are  to  Murray,  Differential 
Equations. 

184.  Definitions.  Classifications.  Solutions.  This  chapter  is 
concerned  with  showing  how  to  obtain  solutions  of  a  few  differe^i- 
tial  equations  which  the  student  is  likely  to  meet  in  elementary 
work  in  mechanics  and  physics. 

Differential  equations  are  equations  that  involve  derivatives  or 
differentials.  Such  equations  have  often  appeared  in  the  preced- 
ing part  of  this  book.      ■  :  '  .?  ■->  -  ^ 

Thus,  in  Art.  37,  Exs.  2,  11,  13,  differential  equations  appear  ;  Equations 
(1),  Art.  60,  (2)-(5),  Art.  67  (a),  (2)-(5),  Art.  67  (c),  (3)-(6),  Art.  67  (d), 
are  differential  equations;  so  also,  in  Art.  68,  are  (I)  a^nd  (2),  Ex.  5  ;  equa- 
tions in  Exs.  13,  14,  and  some  of  the  equations  in  Exs.  10,  11 ;  several  equa- 
tions in  Ex.  1,  Art.  69 ;  Equations  (2)-(4),  Ex.  1,  Art.  73  ;  the  answers  to 
Exs.  2-4,  Art.  73;  in  Ex.  4,  Art.  79;  in  Exs.  5-8,  Art.  80;  Equation  (8), 
Art.  144  ;  etc.,  etc. 

Differential  equations  are  classified  in  the  following  ways,  A 

A.  Differential  equations  are  classified  as  ordinary  differential 
equations  and  partial  differential  equations,  according  as  one,  or 
more  than  one,  independent  variable  is  involved.  Thus,  the  equa- 
tions in  Ex.  4,  Art.  79,  and  in  Exs.  5-8,  Art.  80,  are  partial  differen- 
tial equations ;  the  other  equations  mentioned  above  are  ordinary 
differential  equations.  (Only  ordinary  differential  equations  are 
discussed  in  this  chapter.) 

^B.   Differential  equations  are  classified  as  to  the  order  of  the 
highest  derivative  appearing  in  an  equation.     Thus,  of  the  exam- 
ples cited  above.  Equations  (2)-(5),  Art.  67  (a),  are  equations  of 
the  first  order;  Equations  (2),  Ex.  5,  Art.  68,  and  (8),  Art.  144,  are 
,  r    ,13^4..,  >•. 


184-186.]  DIFFERENTIAL   EQUATIONS.  335 

equations  of  the  second  order;  the  last  equation  but  one  in  Ex.  1, 
Art.  69,  is  an  equation  of  the  nth  order. 

A  solatioii  (or  integral)  of  a  differential  equation  is  a  relation 
between  the  variables  which  satisfies  the   equation.      Thus,  in    p 
Art.  73,  Ex.  1,  relation  (1)  satisfies  Equation .j^4),  and,  accordingly, 
is  a  solution  of  (4).  ^  "      -   4-^-^  ^    ^    ^JT^^  ^^'  ^>^  -cA^^-hfi- 

Ex,  1.    Show  that  relation  (1)  satisfies  Equation  (4)  in  Art.  rSf  Ex.  1. 

Ex.  2.  See  Ex.  4,  Art.  79,  and  Exs.  5-8,  Art.  80.  In  these  examples  the 
equations  in  the  ordinary  functions  are  solutions  of  the  differential  equations 
associated  with  them. 

Ex.  3.  Show  that  the  relations  in  Exs.  2-5,  Art.  73,  are  solutions  of  the 
differential  equations  obtained  in  these  respective  exercises. 

185.  Constants  of  integration.  General  solution.  Particular  solu- 
tions. It  has  been  seen  in  Art.  73,  Ex.  6,  that  the  elimination  of 
n  arbitrary  constants  from  a  relation  between  two  variables  gives 
rise  to  a  differential  equation  of  the  nth  order.  This  suggests  the 
inference  that  the  most  general  solution  of  a  differential  equation 
of  the  nth  order  must  contain  n  arbitrary  constants.  For  a  proof 
of  this,  see  Diff\  Eq.,  Art.  3,  and  Appendix,  Note  C.  Simple 
instances  of  this  principle  have  appeared  in  Art.  73,  Exs.  1-5. 

A  general  solution  of  an  ordinary  differential  equation  is  a  solu- 
tion involving  n  arbitrary  constants.  These  n  constants  are  called 
constants  of  integration.  Particular  solutions  are  obtained  from  the 
general  solution  by  giving  the  arbitrary  constants  of  integration 
particular  values.  The  solutions  of  only  a  few  forms  of  differential 
equations,  even  of  equations  of  the  first  order,  can  be  obtained. 

N.B.  For  a  fuller  treatment  of  the  topics  in  Arts.  184,  185,  see  Diff.  Eq., 
Chap.  I. 

EQUATIONS   OF  THE   FIRST  ORDER. 

186.  Equations  of  the  form  f{x)  dx  -{-F{y)dy  =  0.  Sometimes 
equations  present  themselves  in  this  simple  form,  or  are  readily 
transformable  into  it;  that  is,  to  use  the  expression  commonly 
used,  "  the  variables  are  separable."     The  solution  is  evidently 


^f{x)dx+JF{y)dy^c. 


336  INFINITESIMAL   CALCULUS,  [Ch.  XXI. 

Ex.  1.    Solve  ydx-\-xdy  =  0.  (1) 

On  separating  the  variables,      —  +  ^  zz:  0,  i 

X        y 

and  integrating,  log  x  +  log  y  =  log  c ; 

whence  '  xy  =  c.  (2) 

Solution  (2)  can  be  obtained  directly  from  (1)  on  noting  that  ydx  +  xdy 
is  d(xy). 


Ex.  2.   VI  -  x-^  dy  -i-Vl  -  y^ dx  =  0.      Ex.  3.  n{x  +  a)  dy  +  m(y -\-  b)  dx  =  0. 

187.  Homogeneous  equations.  These  are  equations  of  the  form 
Pclx  ^  Qcly  =  0,  in  which  P  and  Q  are  liomogeneous  functions 
of  the  same  degree  in  x  and  y.  The  substitution  of  voc  for  y 
leads  to  an  equation  in  v  and  x  in  which  the  variables  are  easily 
separable. 

Ex.  1.    (2/2  -  a;2)  dy  +  2xydx  =  0.        Ex.  3.    i/  dx  +  (xy  +  x"^)  dy  =  0. 

Ex.  2.    (a;2  +  ?/2)  dx  +  xydy  =  0.  Ex.  4.    {y^  -  2  xy)  dx  =  {x^  -  2  a:y)  dy. 

188.  Exact  differential  equations.     These  are  equations  of  the 

form 

Fdx+Qdy  =  0,  (1) 

in  which  the  first  member  is  an  exact  differential  (see  Art.  109). 
If  P  and  Q  satisfy  test  (2),  Art.  109,  then  (1)  is  an  exact  differ- 
ential equation,  and  its  solution  is 


C(Pdx-\-Qdy)  =  c. 


Ex.  1.  xdy  +  ydx  =  0.     (See  Ex.  1,  Art.  186.) 

Ex.  2.  (2  xy  +  S)dx-\-  (x^  +  4y)dy  =  0. 

Ex.  3.  (e==  sin  2/  +  2  x)  dx  +  e^  cos  ydy  =  0. 

Ex.  4.  (ax  -  ?/2)  dy  =  (x^  -  ay)  dx. 

Integrating  factors.  Equations  that  are  not  exact  can  be  made 
exact  by  means  of  what  are  termed  integrating  factors.  In  some 
cases  these  factors  are  easily  discoverable. 


187-189.]  DIFFERENTIAL   EQUATIONS,  -   337 

EXAMPLES. 

6.   Solve  xdy  —  ydx  =  0.  (1) 

The  first  member  does  not  satisfy  the  test  in  Art.  109  ;  thus  (1)  is  not  an 
exact  differential  equation.     Multiplication  by  1  ^  xy  gives 

dy_dx_Q, 
y       X 
whence  log  y  —  log  x  =  log  c,  and,  accordingly,  y  =  ex. 
Multiplication  by  1  -^  x'^  gives 

xdy  -y  dx  _  ^  . 

whence  -  =  c,  i.e.  y  =  ex. 

X 

Similarly,  multiplication  by  1  -=-  y'^  makes  (1)  integrable. 
The  multipliers  used  above  are  called  integrating  factors.     In  the  follow- 
ing examples  these  factors  can  be  obtained  by  inspection. 

6.  Solve  (</2  -  x2)  ^y  +  2 xy  dx  =  0.     (See  Ex.  1,  Art.  187.) 
On  rearranging,              y'^  dy  -\-2xydx  —  x^  dy  —  0, 

and  using  the  factor  1  -  y'^  dy  +  ^^Pd^-^^^y  ^  ^ 

yZ 

Whence,  on  mtegration,  y  -\ —  =  c : 

y 

i.e.  x^  -{■  y^  —  cy  z=  0. 

7.  2aydx  =  x(y  —  a)  dy.  8.    (y  -\-  xy^)dx  =  (x^  —  x)dy. 
Note.     On  Integrating  Factors  see  Diff.  Eq.,  Arts.  14-19. 

189.   The  linear  equation    ^^  Py=Q,  (1) 

in  which  P  and  Q  do  not  involve  y.  (It  is  called  linear  because 
the  dependent  variable  and  its  derivative  appear  only  in  the  first 
degree.)  This  is,  perhaps,  the  most  important  equation  of  the 
first  order. 

It  has  been  discovered  that  eJ^**  is  an  integrating  factor  for 
this  equation.     On  using  this  factor, 

e^"'(|+P^)  =  Qj'";  (2) 

whence,  on  integration, 

Note.    For  the  discovery  of  the  integrating  factor,  see  Diff.  Eq.,  Art.  20. 


338  INFINITESIMAL   CALCULUS.  [Ch.  XXI. 

EXAMPLES. 

1.  Show  that  (2)  is  an  exact  differential  equation. 

2.  x^y- -  ay  =  X -^  I. 

dx 

On  using  form  (1),  ^  _  ?  y  ^  1  +  ^-i. 

dx     X 

HereP  =  -^.     .:  ( Pdx  =  - alogx  =  \ogx-''.     .-.  el^'^^  =  x-«„ 
X  J 

On  using  this  factor,  x-'^idy  -  ax-^ dx)  =  x-«(l  +  x-i)  dx; 

and  integrating,  ?/x-«  =  ^^ h  ^—  +  c, 

1  —  a      —  a 

whence  y  =  — ^ +  ex". 

1  —  a      a 

3.  (1  -  x2)  ^--xy=l.  4.    cos^  x  ^^  +  w  =  tan  x. 

c?x  dx 

5.  f?  +  L=^,  =  i; 

dx  X^ 

Some  equations  are  reducible  to  form  (1).     For  example, 

f^+Py=Qy^.  .       (3) 

On  division  by  y^,  y-^  -^  +  Pj/i-"  =  O. 

dx 

On  putting  ?/i-«  =  v,  it  will  be  found  that  (3)  takes  the  linear  form 

|  +  (l-n)Pi;=(l-n)$.  (4) 

6.    Derive  (4)  from  (3). 

7-  ?  +  r^  =  ^^*.  \  8.   ^  =  x^y^-xy. 

dx     1  —  x^  dx 

190.   Equations  not  of  the  first  degree  in  the  derivative.     Three 
types  of  these  equations  will  be  considered  here,  viz.  A,  B,  C,  that 

follow.     (Let  -^  be  denoted  by  p.) 

A.    Equations  reducible  to  the  form  oc  —  f{y,  p).  (1) 

On  taking  the  ^-derivatives,     -=  <f)(  y,  p,  —  \  say.  (2) 

p     V      dyJ 

Possibly,  (2)  may  be  solvg-ble  and  give  a  relation,  say, 

F(p,y,  c)  =  0., (3) 


190.]  DIFFERENTIAL   EQUATIONS.  339 

The  p-eliminant  between  (1)  and  (3)  is  the  solution.  If  this 
eliminant  is  not  easily  obtainable,  Equations  (1)  and  (3),  taken 
together,  may  be  regarded  as  the  solution,  since  particular  corre- 
sponding values  of  x  and  y  can  be  obtained  by  giving  79  particular 
values. 


Ex.  1.                                       x  =  y  +  a  \ogp. 

On  taking  the  ^/-derivative,      1=1+^^;  whence  l-p  = 
P            P  dy 

dy 

On  integrating,                         y  =  c  —  a\og{p  —  \)\ 

d  thence                                    x  =  c  +  a  log  ^  ~    . 

p 

Ex.  2.  phj  -\-2px  =  y.                                           Ex.  Z.   x  = 

-.y+pK 

B.  Equations  reducible  to  the  form  y  =f(x,  p). 

(4) 

On  taking  the  a^derivative,  p  =  <\>lx,  p,  -J-  )  say. 

(6) 

Possibly,  (5)  may  be  solvable  and  give  a  relation,  say, 

F{p,  X,  c)  =  0.  (6) 

The  p-eliminant  between  (4)  and  (6)  is  the  required  solution. 
If  this  eliminant  is  not  easily  obtainable,  Equations  (4)  and  (6), 
taken  together,  may  be  regarded  as  the  solution,  since  they  suffice 
for  the  determination  of  x  and  y  by  assigning  values  to  a  param- 
eter p. 

Ex.  4.    Ay  =  x^  +  p2.  Ex.  5.    2y-\-p'i  =  2 x^ 

C.  Clairaut's  equation,  viz.  y  =poc -\- f{p).  (7) 

In  this  case  y  =  cx  -\-f(c)  (8) 

is  obviously  a  solution. 

This  solution  can  be  obtained  on  treating  (7)  like  (4),  of  which  it  is  a 
special  case. 

Thus,  on  taking  the  2c-derivatives  in  (7), 

dp 


p=p-]-[x+f'(p)-]^^ 


dp 


From  this,         x+f'(p)  =  0    (9),  or  ^^^'  ^^^^ 

Equation  (10)  gives  p  =  c. 

Substitution  of  this'in  (7)  gives  (8). 

As  to  the  part  played  by  (9)  see  Diff.  Eq.^  Art.  34. 


340  INFINITESIMAL   CALCULUS.  [Ch.  XXI. 

EXAMPLES. 


a 


6.   y=px-\--'  T.  y  =px  +  aVl  -h  p'^. 

8.  x'^(y  —  px)  =  yp'^.     [Suggestion :  Put  x^  =  u,  y^  =  v.'] 

Note  1.  Sometimes  the  first  member  of  an  equation  f(x,  y,  p)  =0  is 
resolvable  into  factors.  In  such  a  case  equate  each  factor  to  zero,  and  solve 
the  equation  thus  made.  (This  is  analogous  to  the  method  pursued  in  solv- 
ing rational  algebraic  equations  involving  one  unknown.) 

9.  Solve  p^  -  p'^(x  +  y  -\-  2)  -{-  p(xy  -\-  2  x  +  2  y)  -  2  xy  =  a. 
On  factoring,  (p  —  x)  =  0,  p  —  y  =  0,  p  —  2  =  0. 

On  solving,  2y  =  x^  -{■  c,  y  =  ce*,  y  =  2  x  -\-  c. 

These  solutions  may  be  combined  together, 

(2y  -x^  -c)(y  -  ce^) (y  -2x-c)=0. 

Note  2,  On  Equations  of  the  first  order  which  are  not  of  the  first  degree 
see  Diff.  Eq. ,  Chap.  III. 

191.  Singular  solutions.  Let  a  differential  equation  f(x,  y,  p)=0 
have  a  solution  f{x,  y,  c)  =  0.  The  latter  is  geometrically  repre- 
sented by  a  family  of  curves.  The  equation  of  the  envelope  of 
this  family  (Art.  154)  is  termed  the  singular  solution  of  the  differ- 
ential equation.  That  the  equation  of  the  envelope  is  a  solution 
is  evident  from  the  definition  of  an  envelope  (see  Art.  154)  and 
this  fact,  viz.  that  at  any  point  on  any  one  of  the  curves  of  the 
family  the  coordinates  of  the  point  and  the  slope  of  the  curve 
satisfy  the  differential  equation.  The  singular  sohition  is  obviously 
distinct  from  the  general  solution  and  from  any  particular  solution. 

For  example,  the  general  solution  [(8),  Art.  190]  of  Clairaut's  equation 
is,  geometrically,  a  family  of  straight  lines.  The  envelope  of  this  family  of 
lines  is  the  singular  solution  of  (7).  The  envelope  of  (8)  may  be  obtained 
by  the  method  shown  in  Art.  157.  Differentiation  of  the  members  of  (8) 
with  respect  to  c  gives  0  —  x  A-  f'Cc') 

The  envelope  is  the  c-eliminant  between  this  equation  and  (8). 

EXAMPLES. 

1.  Show  that  the  singular  solution  of  Ex.  6,  Art.  190,  is  y"^  =  4  ax. 

2.  Find  the  singular  solutions  of  the  equations  in  Exs.  7,  8,  Art.  190. 


191,  192.] 


DIFFERENTIAL  EQ UA  TIONS. 


Ul 


3.    Find  the  general  solution  and  the  singular  solution  of: 

(1)  y=px+p'^.         (2)  p'^x  =  y.        (3)  %  a(\ -\- p)^  =  21{x  +  y)(\  - p^. 

Note  1.  The  singular  solution  can  a-lso  be  derived  directly  from  the  dif- 
ferential equation,  without  finding  the  general  solution  ;  see  reference  below. 

Note  2.    On  Singular  Solutions  see  Diff.  Eq.,  Chap.  IV.,  pages  40-49. 

192.  Orthogonal  Trajectories.  Associated  with  a  family  of  curves 
(Art.  154),  there  may  be  another  family  whose  members  intersect 
the  members  of  the  first  family  at  right  angles.  An  instance  is 
given  in  Ex.  1.  The  members  of  the  one  family  are  said  to  be 
orthogonal  trajectories  of  the  other  family. 

For  example,  the  orthogonal  trajectories  of  a  family  of  concentric  circles 
are  the  straight  lines  passing  through  the  common  centre  of  the  circles. 


A,    To  find  the  orthogonal  trajectories  of  the  family 


(1) 


in  which  a  is  the  arbitrary  parameter.  Let  the  differential 
equation  of  this  family,  which  is  obtained  by  the  elimination  of 
a  (see  Art.  73),  be  ,  ,,  ^ 

*{<^,y,^)  =  o.  (2) 


Fig.  102. 


Fig.  103. 


Let  P  be  any  point,  through  which  pass  a  curve  of  the  family 
and  an  orthogonal  trajectory  of  the  family,  as  shown  in  Fig.  102. 
For  the  moment,  for  the  sake  of  distinction,  let  (x,  y)  denote  the 
coordinates  of  P  regarded  as  a  point  on  the  given  curve,  and  let 


342  INFINITESIMAL   CALCULUS.  [Ch.  XXI. 

(X,  Y)  denote  the  coordinates  of  P  regarded  as  a  point  on  the 
trajectory.     At  P  the  slope  of  the  tangent  to  the  curve  and  the 

slope  of  the  tangent  to  the  trajectory  are  respectively  —  and  — ;:• 

Since  these  tangents  are  at  right  angles  to  each  other, 


dy  _     dX 
dx         dY 

Also                                   x=  X,  and  y  = 

Y 

Substitution  in  (2)  gives 

But  P(X,  Y)  is  any  point  on  any  trajectory.     Accordingly,  (3) 
or,  what  is  the  same  equation, 

is  the  differential  equation  of  the  orthogonal  trajectories  of  the 
curves  (1)  or  (2). 

Hence:   To  find  the  differential  equation  of  the  family  of  orthog- 

dx         dv 
onal  trajectories  of  a  given  family  of  curves  substitute for  -^ 

in  the  differential  equation  of  the  given  family.  ^ 


EXAMPLES. 

1.    Find  the  orthogonal  trajectories  of  the  family  of  circles  which  pass 
through  the  origin  and  have  their  centres  on  the  x-axis. 
The  equation  of  these  circles  is 

x2  +  ?/  =  2  ax,  (4) 

in  which  a  is  the  arbitrary  parameter. 

On  differentiation  and  the  elimination  of  a  (Art.  73),  there  is  obtained 
the  differential  equation  of  the  family,  viz. 

y2_x'^_2xy^  =  0.  (5) 

dx 

The  substitution  of    -  —    for  ^  gives  the  differential  equation  of  the 
orthogonal  curves,  viz.  •=' 

y2_r^2^2XV~=0.  (6) 


192.] 


DIFFERENTIAL  EQ UA  TIONS. 
T 


343 


Fig.  104. 
Integration  of  (6)  [see  Art.  188,  Ex.  6]  gives 

a;2  _|.  2,2  —  cy, 


(7) 


the  ortliogonal  family,  viz.  a  family  of  circles  passing  through  the  origin  and 
having  their  centres  on  the  y-axis.     (See  Fig.  104.) 

2.  Obtain  the  orthogonal  trajectories  of  the  circles  (7),  viz.  the  circles  (4). 

3.  Derive  the  equation  of  the  orthogonal  trajectories  of  the  family  of 

lines  y  =  mx. 

4.  Derive  the  equation  of  the  family  of  concentric  circles  whose  centre 
is  at  the  origin. 


B.  To  find  the  orthogonal  trajectories  of  the  family 


(8) 


in  which  c  is  the  arbitrary  parameter.     Let  the  differential  equa- 
tion of  this  family,  which  is  obtained  by  the  elimination  of  c,  be 


l^^r,  0, 


dr 
d9 


0. 


(9) 


344  INFINITESIMAL   CALCULUS.  [Ch.  XXI. 

Let  P  be  any  point  through  which  pass  a  curve  of  the  given 
family  and  an  orthogonal  trajectory  of  the  family,  as  shown  in  Fig. 
103.  For  the  moment,  for  the  sake  of  distinction,  let  (r,  $)  denote 
the  coordinates  of  P  regarded  as  a  point  on  the  given  curve,  and 
let  {R,  ©)  denote  the  coordinates  of  P  regarded  as  a  point  on  the 
trajectory.  At  P  (see  Art.  60)  the  tangent  to  the  given  curve  and 
the  tangent  to  the  trajectory  make  with  the  radius  vector  angles 

whose  tansrents  are  respectively  r—  and  R^- 
""  ^  ^     dr  dR 

Since  these  tangent  lines  are  at  right  angles  to  each  other, 

dO  1  ,  dr  „  d®  „9  d® 

r  —  = 7— :  whence  —  =  —  rR  —  =  —  R-  -— - 

dr  j^d®'  dd  dR  dR 

dR 

Accordingly  (9)  may  be  written 

But  P{R,  0)  is  any  point  on  any  trajectory.  Accordingly  (10), 
or  the  same  expression  in  the  usual  symbols  r  and  6, 

Jf  (»•>»> -'•'f:)  =  o,  (10') 

is  the  differential  equation  of  the  orthogonal  trajectories  of  the 
curves  (8)  or  (9). 

Hence  :  To  find  the  differential  equation  of  the  family  of  orthogo- 
nal trajectories  of  a  given  family  of  curves,  substitute  —i^—  for  — 
in  the  differential  equation  of  the  given  family. 


EXAMPLES. 

5.    Find  the  orthogonal  trajectories  of  the  set  of  circles  r  =  a  cos  ^,  a 

being  the  parameter. 

Differentiation  and  the  elimination  of  a  gives  the  differential  equation  of 

these  circles,  viz.  .i^ 

—  +  r  tan  (9  =  0. 
dd 


On  substituting,  as  directed  above,  there  is  obtained 


r^  =  tan 


dr 

the  differential  equation  of  the  orthogonal  trajectories.      Integration  gives 
another  family  of  circles  r  =  c  sin  ^.  (11) 


192.]  DIFFERENTIAL   EQUATIONS.  345 

6.  Sketch  the  families  of  circles  in  Ex,  5,  and  show  that  the  problem 
and  result  in  Ex.  5  are  practically  the  same  as  the  problem  and  result  in  Ex.  1. 

7.  Find  the  orthogonal  trajectories  of  circles  (11),  viz.  the  circles  in 
Ex.5. 

N.B.  Various  geometrical  problems  requiring  differential  equations  are 
given  in  the  following  examples. 

Note  1.  On  applications  of  differential  equations  of  the  first  order,  see 
Diff.  Eq.,  Chap.  V. 

8.  Find  the  curves  respectively  orthogonal  to   each   of   the   following 

families  of  curves  (sketch  the  curves  and  their  trajectories)  :   (1)  the  parabolas 

1/2  =  4 ax;  (2)  the  hyperbolas  xy  =  ^•2  ;   (3)  the  curves  a"-iy  =  a;«;  interpret 

the  cases  n  =  0,  1,  -  1,  2,  -  2,  ±  i,  ±  |,  respectively ;  (4)  the  hypocycloids 

2        1^ 
x^  -{-  y^  =  a^  ;  (5)  the  parabolas  y  =  ax'^;  (6)  the  cardioids  r  =  a(l-cosd); 

(7)  the  curves  r" sin  nd  =  a"*;  (8)  the  curves  r^  =  a" cos  n6 ;  (9)  the  lemnis- 

cates  r2  =  a2  cos  2  ^ ;  (10)  the  confocal  and  coaxial  parabolas  r  = —  ; 

1-1-  cos  0 
(11)  the  circles  x^  +  y^  +  2my  =  a^,  in  which  m  is  the  parameter.        ^ 

9.  (a)  Show  that  the  differential  equation  of  the  confocal  parabolas 
y"^  =  4  a(x  -\-  a)  is  the  same  as  the  differential  equation  of  the  orthogonal 
curves,  and  interpret  the  result.     (&)  Show  that  the  differential  equation  of 

the  confocal  conies  — 1 ^^  =  1  is  the  same  as  the  differential  equation 

a2  +  z      6-2  _|.  2 

of  the  orthogonal  curves,  and  interpret  the  result. 

10.  Find  the  curve  such  that  the  product  of  the  lengths  of  the  perpen- 
diculars drawn  from  two  fixed  points  to  any  tangent  is  constant. 

11.  Find  the  curve  such  that  the  product  of  the  lengths  of  the  perpen- 
diculars drawn  from  two  fixed  points  to  any  normal  is  constant. 

12.  Find  the  curve  such  that  the  tangent  intercepts  on  the  perpendiculars 
to  the  axis  of  x  at  the  points  (a,  0),  (-a,  0),  lengths  whose  product  is  b^. 

13.  Find  the  curve  such  that  the  product  of  the  lengths  of  the  intercepts 
made  by  any  tangent  on  the  coordinate  axes,  is  equal  to  a  constant  a^. 

14.  Find  the  curve  such  that  the  sum  of  the  intercepts  made  by  any 
tangent  on  the  coordinate  axes  is  equal  to  a  constant  a. 

EQUATIONS   OF   THE   SECOND   AND   HIGHER   ORDERS. 

Only  a  very  few  classes  of  these  equations  will  be  solved  here ; 
namely,  simple  forms  of  linear  equations  with  constant  coefficients 
and  homogeneous  linear  equations.  Three  special  equations  of 
the  second  order  will  also  be  briefly  discussed. 


346  INFINITESIMAL   CALCULUS.  [Ch.  XXI. 

193.    Linear  Equations.     Linear  equations  are  those  which  are' 
of  the  first  degree  in  the  dependent  variable  and  its  derivatives. 
The  general  type  of  these  equations  is 

in  which  Pj,  P,,  ••-,  P„,  X,  do  not  involve  y  or  its  derivatives. 

(For  some  general  properties  of  these  equations  see  Murray,  Integral 
Calculus,  Art.  113,  Diff.  Eq.,  Art.  49.) 

A,  The  linear  equation ^^^+ Pi ^^ ^+P2^^^ ^^/^.....^p.  2,^0  (1) 

die"  dx''-^         dx'"^-'^ 

in  which  the  coefficients  Pi,P2,  •••,  P„,  are  constants. 

The  substitution  of  e^""  for  y  in  the  first  member,  gives 
(m"  +  P-^m''-^  +  P.:{m''-'-  +  •••  +  P„)e'"^ 

This  expression  is  zero  for  all  values  of  m  that  satisfy  the 
equation  ^.  ^  p^^^^^-x  ^  p,^^n-2  _^  . . .  _^  7.^  _  0 ;  (2) 

and,  accordingly,  for  each  of  these  values  of  m,  y  =  e"^""  is  a  solu- 
tion of  (1).  Equation  (2)  is  called  the  auxiliary  equation.  Let 
mi,  mg,  •••,  m„,  be  its  roots.  Substitution  will  show  that  y  =  qe'"!'', 
y  =  C2e™2*,  ...,?/  =  c,,e'^n'=,  and  also 

y  =  de"*!^  +  C2e™2^  +  •  •  •  +  c„e"'"'',  (3) 

in  which  the  c's  are  arbitrary  constants,  are  solutions  of  (1). 
Solution  (3)  contains  n  arbitrary  constants  and,  accordingly,  is  the 
general  solution. 

Note  1.    If  two  roots   of^^  (2)   are  imaginary,  say  a  +  i/3  and  a  —  i/S,  z 
denoting  V—  1,  the  correspon^Jing  sohition  is  , 

According  to  Art.  179  this  may  be  put  in  the  form 

y  =  ea^(cie''^^  +  026^'^*) 

=  ea=^{ci(cos  ^x  4-  i  sin  j8a:)  +  C2(cos  /3x  —  i  sin  jSa;)}, 

=  e^^^lCci  4-  C2)  cos  /3x  +  i(ci  —  C2)  sin  jSa;}, 

=  ea=^(^  cos  j8x  +  -S  sin  /3a:) , 

in  which  A  and  5  are  arbitrary  constants,  since  ci  and  C2  are  arbitrary 
constants. 


193.]  DIFFERENTIAL   EQUATIONS.  347 

^     Note  2.    If  ttoo  roots  of  (2)  are  equal,  say  nit  and  m^  each  equal  to  a,  the 
corresponding  solution ,  viz. 

becomes  ?/  3=  (ci  +  C2)e«*,  i.e.  y  =  ce"*, 

which  does  not  involve  two  arbitrary  constants.     Put  mo  =  a  -h  h  ;  then  the 

solution  takes  the  form  ,  ^^, 

^  y  =  Cie"^  +  C2e(''+*)=', 

=  e«'(ci  +  626*=^). 

On  expanding  e'"'  in  the  exponential  series  (Art.  178,  Ex.  7),  this  equation 
becomes 

y  =  e«'^(^l  -|-  J5x  +  ^  C2h^x!^  +  terms  in  ascending  powers  of  h),  (4) 

in  which  A  =  ci  +  Co  and  B  —  c^h.     On  letting  /i  approach  zero  in  (4),  the 

latter  becomes  \  ,  ^       ^  x 

\  y  =  e'"{A-^  Bx). 

(The  numbers  Ci  and  co  can  always  be  chosen  so  that  ci  +  C2  and  C2^  are 
finite.) 

If  a  root  a  of  (2)  is  repeated  r  times,  the  corresponding  solution  is 

2/  =  (ci  +  C2X  +  csx^  -f  ...  +  Cra;''"^)e"'. 

Note  3.     On  Equation  (1),  see  Diff.  Eq.,  Arts.  50-55. 

EXAMPLES. 

1.  Solve  ^^-3^^  +  2y  =  0. 

dx^        dx 

The  auxiliary  equation  is     m^  —  3  w  +  2  =  0  ; 
its  roots  are  —  2,  1,  1. 

Accordingly,  the  solution  is        y  =  Cie~'^  +  (C2  +  Czx)^'. 

2.  Solve  ^+a2y  zzO. 

The  auxiliary  equation  is   iii^  +  a^  =  O; 
its  roots  are  ai,   —  ai. 

Accordingly,  its  solution  is        y  =  cie"'^  +  C2e"«** 

=  ^  cos  «x  +  -S  sin  ax.     (See  Ex.  1,  Art.  73.) 

3.  Solve  the  following  differential  equations  : 

(1)  D^y  -  4  Dy-^  13  y  =  0.      (2)  D^y -1  Dy  +  Qy  =  0. 

(3)  ^_12^i^-162/  =  0.       (4)  ^-10^  +  62^-160«^  +  136y  =  0. 


348  INFINITESIMAL   CALCULUS,  [Ch.  XXI. 

B,  The  "homogeneous"  linear  equation  i 

m  ^t'/w*c/i  pi,  P2,  '",  Pn,  cire  constants. 

First  method  of  solution.  If  the  independent  variable  x  be 
changed  to  z  by  means  of  the  relation 

z  =  log  X,  i.e.  X  =  e', 

the  equation  will  be  transformed  into  an  equation  with  constant 
coefficients.  (For  examples,  see  Art.  92  and  Exs.  3  (i),  (v),  (vi), 
page  147^ 

4.  Show  the  truth  of  the  statement  last  made. 

5.  Solve  Exs.  7  below  by  this  method. 

Second  metliod  of  solution.  The  substitution  of  x""  for  y  in  the 
first  member  of  equation  (5)  gives 

[iftiim  —  V)'"(jn~-n-\-V)  -{-p^m{m  —  1)  •  •  •  (m  —  w  -f  2)  H 1-Ph]-^"*- 

This  is  zero  for  all  values  of  m  that  satisfy  the  equation 

m(m  — l)«--(m— n  +  l)H-j9im(m— 1)  '••{m,  —  n-\-2)-\ \-p^=z{).     (6) 

Let  the  roots  of  (6)  be  m^  m^,  •••,  m„;  then  it  can  be  shown, 
as  in  the  case  of  solution  (3)  and  equation  (1),  that 

y  =  CiX"^'  -f-  C2iC'"2  _| _j_  c^Qfh, 

is  the  general  solution  of  equation  (5). 

The  forms  of  this  solution,  when  the  auxiliary  equation  (6) 
has  repeated  roots  or  imaginary  roots,  will  become  apparent  on 
solving  equation  (5)  by  the  first  method. 

EXAMPLES. 

6.  Show  that  the  solution  of  (5)  corresponding  to  an  r-tuple  root  m  of 

(6),  is  ?/  =  a;'»[ci  + C2loga;  + C3(logx)2-|- ...  +  CrClogx)*— 1]  ;  and  show  that 
the  solution  of  (5)  corresponding  to  two  imaginary  roots  a  + 1/3,  a  —  i^,  of  (6) ,  is 

y  =  x'^[ci  cos  (j8  log  x)  +  C2  sin  (/S  log  a;)]. 


194.]  DIFFERENTIAL   EQUATIONS.  349 

7.  Solve  the  following  equations  : 

(1)  x^D^y  -  xDy  -h2y  =  0.  (2)  x'^D'^y  -  xDy  +  y  =  0. 

(3)  x'^D-y  -3xDy-\-^y  =  0.  (4)  x^D^y  +  2  x^^D^y -\- 2  y  =  0. 

Note  3.     Equations  of  the  form 

are  reducible  to  the  homogeneous  linear  form,  by  putting  a  -{-  bx  =  z. 

8.  Show  the  truth  of  the  last  statement. 

9.  Solve  (5  +  2  a;)2^- 0(5  +  2  a;)^+ 8  2/  =  0. 

Note  4.     On  Equation  (5),  see  Diff.  Eq.,  Arts.  65,  66,  71. 
194.     Special  equations  of  the  second  order. 

A,  Equations  of  the  form  -j-^^  =  f(y)» 

dy 
For  these  equations  2  y-  is  an  integrating  factor. 

EXAMPLES. 

1.  ^  +  a^i/  =  0.     (See  Ex.  2,  Art.  193.) 

On  integrating,  [y  j   =  -  aV  +  ^ 

=  a2(c2  —  y-)^  on  putting  aV  for  k. 

On  separating  the  variables,         '       =  adx, 
y/c^  —  y'^ 

V 
and  integrating,  sin-i-  =  ax  ■}-  a. 

This  result  may  be  written  y  =  c  sin  (ax  +  a) , 

or  y  =  Asinax  +  B  cos  ax. 

2.  Show  the  equivalence  of  the  last  two  forms.     Express  A  and  jB.in 
terms  of  c  and  a,  and  express  c  and  a  in  terms  of  A  and  J5. 

3;   Show  that  2  y  is  an  integrating  factor  in  case  A. 

C  Solve  the  following  equations  : 

(3)  If  ^2  ~  ~2~'  ^'^^  ^'  S^^^^  ^^^^t  1~~^  ^^^  X  =  a,  when  t  =  0. 

(tt  8  ,  (It 


350  INFINITESIMAL   CALCULUS,  [Ch.  XXI. 

B,  Equations  of  the  form  /(|^,  il'  ^)  "  ^-  (1) 

On  letting  j>  denote  — ,  this  may  be  written  /(  -^,  p,  x\=  0.  (2) 
dx  \dx         J 

Integration  of  (2)  may  give     <ji{p,  x,  c)  =  0, 
and  this  may  happen  to  be  integrable. 

EXAMPLES. 

5.  Find  the  curve  whose  radius  of  curvature  is  constant  and  equal  to  a. 
(This  example  is  the  converse  of  Art.  147.) 

6.  Solve  the  following  equations  : 

(2)  xD-'y  +  Dy  =  0.  (4)  (1  +  x)D'-y  +  Dy  +  x  =  0. 

C.  Equations  of  tlie  form  /(|^,  ^|,  y)  =0,  (1) 

This  (see  Art.  90)  may  be  written 

f(p'^,P,y)  =  o.  (2) 


dy 
Integration  of  (2)  may  give 

F(p,  y,  c)  =  0, 
and  this  may  happen  to  be  integrable. 

EXAMPLES. 

7.  Solve  S  +  ^^^  "^  ^-     ^^^^  ^^*  ^'^ 

This  is  p^=z  —  a^y.  • 

dy 
Now  proceed  as  in  Ex.  1. 

8.  Solve  the  following  equations  : 

(3)  y^D^y  +  1  =  0.  (4)  Dhj  +  (Dy)''  +  1=0. 

Note  5.     For  the  solution  of  equations  in  the  form  D^^y  =  f(x)f  see 
Art.  129. 


194.]  DIFFERENTIAL  EQUATIONS.  351 

Note  6.     On  forms  like  A,  B,  C,  see  Diff.  Eq.,  Arts.  77,  78,  79,  respectively. 

Note  7.  References  for  collateral  reading.  For  a  brief  treatment  of 
differential  equations  and  for  interesting  practical  examples,  see  Lamb,  Cal- 
culus, Chaps.  XI.,  XII.  (pp.  456-540)  ;  also  see  F.  G.  Taylor,  Calculus, 
Chaps.  XXIX.-XXXIV.  (pp.  493-564),  and  Gibson,  Calculus,  Chap.  XX. 
(pp.  424-441). 

EXAMPLES. 

Solve  the  following  equations  : 
(1)  rdd  =  tan  6  dr.     (2)   (1  +  y)dx  +  x(x  +  y)dy  =  0. 
J^S)  (iy-\-3x)dyi-(y-2x)dx=0.    (4)  x^-y=V^^+^'.    (5)  ^+y tana:=l. 

(6)  ic  ~  -  2  ?/  =  .^4  vTT^.      (7)   (6  ic  +  4  y  +  5)dx  -\-{lO  y  +  4x  +  l)dy  =  0. 

dy        ix 


(8)  y(y  dx  -  X  dy)  +  xVx^  +  y^  dy  =  0.      (9)  ^  + -^—^  y  =  ^-^-_^^. 

^^^^  '^fx""  ^^  ^V^'      (11)  ^  -  yi^  =  «i>'-      (12)  2/-2  =  a2(i+^2). 

{IZ)  {px-y){py -\-x)=h'^p.      {\A)  pH^ -^  x'^py  =  \.      {\b)x  =  2y-^p\ 

(16)  p^  4-  2py  cotx  =  2/2,     (17)  yy/i  Jf-  p-i  ~  a;  also  find  the  singular  solution. 

(18)  y  —  px  =  Vb'^  +  a^p^^ ;  also  find  the  singular  solution.    (19)  xp^  =(x  —  a)'^, 

d^ii  d^v 

and  also  find  the  singular  solution.     (20)  -jf^  —  a*y  =  0.     (21)  —^  +  4  ?/  =  0- 

(.24)  x^g  +  3.^g  +  .|f +  .  =  0.     (25)  (.  +  a)^g-4(x+a)2+6,=0. 

^^=-  ^^m=h  ^^^^-  <-)(S)=«(i)- 


APPENDIX. 
NOTE   A. 

ON   HYPERBOLIC    FUNCTIONS. 

1.  This  note  gives  a  short  account  of  hyperbolic  functions  and 
their  properties.  The  student  will  probably  meet  these  functions 
in  his  reading ;  for  many  results  in  pure  and  applied  mathematics 
can  be  expressed  in  terms  of  them,  and  their  values  are  tabulated 
for  certain  ranges  of  numbers.*  There  are  close  analogies  between 
the  hyperbolic  functions  and  the  circular  (or  trigonometric)  func- 
tions (a)  in  their  algebraic  definitions,  (b)  in  their  connection  with 
certain  integrals,  (c)  in  their  respective  relations  to  the  rectangular 
hyperbola  and  the  circle. 

2.  Names,  symbols,  and  algebraic  definitions  of  the  hyperbolic 
functions.  The  hyperbolic  functions  of  a  number  x  are  its  hyper- 
bolic sine,  hyperbolic  cosine,  hyperbolic  tangent,  •••,  hyperbolic 
cosecant,  and  the  corresponding  six  inverse  functions.  These  func- 
tions have  been  respectively  denoted  by  the  symbols  sink  x,  cosh  x, 
tanlix,  cotlix,  sechx,  cosechx,  sinh'^x,  etc.  These  are  the  symbols 
in  common  use.  As  to  symbols  for  the  hyperbolic  functions,  the 
following  suggestion  has  been  made  by  Professor  George  M. 
Minchin  in  Nature,  Vol.  65  (April  10,  1902),  page  531:  "If  the 
prefix  7iy  were  put  to  each  of  the  trigonometrical  functions,  all  the 
names  would  be  pronounceable  and  not  too  long.  Thus,  Jiysin  x, 
hytanx,  etc.,  would  at  once  be  pronounceable  and  indicate  the 

*  See  tables  of  the  hyperbolic  functions  of  numbers  in  Peirce,  Short  Table 
of  Integrals  (revised  edition,  1902),  pages  120-123  ;  Lamb,  Calculus,  Table 
E,  page  611  ;  Merriman  and  Woodward,  Higher  Mathematics,  pages  162-168. 

353 


354  INFINITESIMAL   CALCULUS. 

hyperbolic   nature   of   the    functions."      This    notation   will   be 
adopted  in  this  note.* 

The  direct  hyperbolic  functions  are  algebraically  defined  as  follows : 

hysin  oc  =  - — -^ — >  hycos  dc  =       ^  — 5 

hytan^=^y^^JL^^^"-^"",      hycot^  ^  |^^^?^^  ^^"  +  ^"%     (1) 
hycos  a?     e^ -\- e-i>^  hysma?     e^  -  e-^      ^^ 

hysec  a?  = >  hycosec  a? 


hycos  a?  hysin  a? 

There  is  evidently  a  close  analogy  between  these  definitions 
and  the  definitions  and  properties  of  the  circular  functions.  [See 
the  exponential  expressions  (or  definitions)  for  sin  x  and  cos  x  in 
Art.  179.] 

From  the  definitions  for  hysin  x  and  hycos  x  can  be  deduced,  by 
means  of  the  expansions  for  e"'  and  6""=  (see  Art.  178,  Ex.  7),  the 
following  series,  which  are  analogous  to  the  series  for  sino?  and 
cos  x  (Art.  178,  Exs.  2,  5) : 


hysin  a3  =  a? +  1^  +  1^  +  .... 
hycos  a.  =  l+|^  +  |*  +  ...5 


(2) 


The  second  members  in  equations  (2)  may  be  regarded  as  defi- 
nitions of  hysin  x  and  hycos  x. 

EXAMPLES. 

1.  Derive  the  following  relations,  both  from  the  exponential  defini- 
tions of  sin  aj,  cos  a;,  hysin  x,  hycos  x,  and  from  the  expansions  of  these  func- 
tions in  series  :  (1)  cos  x  =  hycos  (ix) ;  (2)  i  sinx  =  hysin  {ix)  ;  (3)  cos  {ix) 
=  hycos  x  ;  (4)  sin  (ix)  =  i  hysin  x. 

2.  (a)  Show  that  e^  =  hycos  x  +  hysin  x,  e-^  =  hycos  x  —  hysin  x. 
[Compare  Art.  179  (1),  (2).]  (&)  Show  that  hysinO  =  0,  hycosO  =  l, 
hy tan  0  =  0,  hysin  qo  =  00,  hycos  qo  =  qo,  hytan  00  =  1,  hysin  (—x)  = 
—  hysin  x,  hycos  (  —  x)  =  hycos  x,  hytan  (  —  x)  =  —  hytan  x. 

*  The  symbols  used  in  W.  B.  Smith's  Infinitesimal  Analysis  are  hs,  he, 
ht,  hct,  hsf,  hcsc. 


APPENDIX,  355 

3.  Show  that  the  following  relations  exist  between  the  hyperbolic 
functions  : 

(1)  hycos^x  —  hysin2x  =  1 ;  (2)  hysec^x  +  hytan^x  =  1  ; 

(3)  hysin  {x  ±  y)=  hysin  x  •  hycos  y  ±  hycos  x  ■  hysin  y  ; 

(4)  hycos  (x  ±y)—  hycos  x  •  hycos  y  ±  hysin  x  •  hysin  y  ; 

(5)  hytan  (x  ±  y)  =  (hytan  x  ±  hytan  «/)  -^  (1  ±  hytan  x  •  hytan  y)  ; 

(6)  hysin  2  a;  =  2  hysin  x  •  hycos  x  ; 

(7)  hycos 2x  =  hycos^ x  +  hysin^ x  =  2  hycos^ x  —  1  =  1  +  2  hysin^ x  ; 

(8)  hytan  2  x  =  2  hytan  x  -^(1  +  hytan^  x). 

Compare  these  relations  with  the  corresponding  relations  between  the 
circular  functions. 

4.  Show    the    following:     (1)  ^^^^^^^5^  =  hycos  x ;     (2)  ^(hycosx)^ 

dx  dx 

hysin  x;  (3)  ^^(M^^^hysec^x  ;  (4)  ^^^y^^^^)^ -hycsc^x;  (5)  ^jhysecx) 

dx  dx  dx 

=  —  hysec  x  •  hytan  x ;  (6)     C  ycsca;;  _  _  j^y^^g^  ^  ,  hycot  x  ;  (7)  i  hysin  x  dx 

r  ^  r 

=  hycos  x;      (8)    |  hycos  x  dx  =  hysin  x  ;      (9)    |  hytan  x  rfx  =  log  (hycos  x)  ; 

(10)    \  hycot  xdx  =  log  (hysin  x)  ;  (H)    l  hysec  XfZx  =  2  tan-^e^  ; 

(12)    i  hy  esc  xfZx  =  log  (hytan -j.     Compare  these  relations  with   the  cor- 
responding relations  between  the  circular  functions. 

5.  Make  graphs  of  the  functions  hysin  x,  hycos  x,  hytan  x.  (See  Lamb, 
Calculus^  pp.  42,  43.) 

y  X  X 

6.  Show  that  the  slope  of  the  catenary  -  =  hycos-  is  hysin--  Sketch 
^,  .  ""    a  a  a 

this  curve. 

Inrerse  hyperbolic  functions.  The  statement  "the  hyperbolic 
sine  of  y  is  ic"  is  equivalent  to  the  statement  "?/  is  a  number 
whose  hyperbolic  sine  is  ic."  These  statements  are  expressed  in 
mathematical  shorthand, 

hysin  y  =  oc,    y  =  hysin-i  ac.  (3) 

The  last  symbol  is  read  "  the  inverse  hyperbolic  sine  of  a;,"  or 
"the  anti-hyperbolic  sine  of  x^  The  other  inverse  hyperbolic 
functions  are  defined  and  symbolised  in  a  similar  manner. 

The  inverse  hyperbolic  functions  can  also  be  expressed  in  terms 
of  logarithmic  functions,  and  thus  they  may  be  given  logarithmic 
definitions.  (This  might  have  been  expected,  for  the  direct  hyper- 
bolic functions  are  defined  in  terms  of  exponential  functions,  and 
the  logarithm  is  the  inverse  of  the  exponential.) 


5Q  INFINITESIMAL  CALCULUS. 

Let  hysin?/  =  ic;    then  a;  =  ^(e^  —  e"*'). 

This  equation  reduces  to  e-^  —  2  ice^  —  1  =  0. 


On  solving  for  e^,         ^  =  x -\-  Vjtf  +  1.  (4) 

(For  real  values  of  y,  e^  being  positive,  the  positive  sign  of  the 
radical  must  be  taken.) 


From  (4)  y  =  hysin-i  oc  =  log(x  +  va?2  +  i),  (5) 

N.B.     The  base  of  the  logarithms  in  this  note  is  e. 

In  a  similar  manner,  on  putting 

X  =  hycos  ?/  =  1  (e^  -f  e-^)j 
and  solving  for  e^, 


e^  =  x±V^- 1.  (6) 

For  real  values  of  y,  x  is  greater  than  1  and  both  signs  of  the 
radical  can  be  taken. 

From  (6)  and  the  fact  that  (x  +  V^  —  l)(x—  ^xr  —  1)  =  1,  and 
thus  log  (x  —  Va^  —  1)  =  —  log  (x  -f  -\/otf  —  1),  it  follows  that 


y  =  hycos-i  x  =  ±  \og(ac  +  V^a  -  l) .  (7) 

In  a  similar  manner  it  can  be  shown  that 

hytaB-ia,  =  |logl±|,  (8) 

where  a^  <  1  for  real  values  of  hytan~^  x ;  and  that 

hycot-ix=|log|^,  (9) 

where  a:^  >  1  for  real  values  of  hycot"^  x. 


EXAMPLES. 

7.  Derive  the  relations  (7),  (8),  (9). 

8.  Solve  equations  (5),    (7),    (8),    (9),  for  x  in  terms  of  y,  and   thus 
obtain  the  definitions  of  the  direct  hyperbolic  functions. 


9.    Show  that  the  differentials  of  hysin-i  x,  hycos~i  x,  hytan-i  x,  hycot-^  x, 

are  respectively        ^^      ,  ±       ^^      .  -^-  for  x2  <  1, ^-  for  x2  >  1. 

Vx^  +  1        Vx^  -  1   1  -  a;2  x2  -  1 

Compare  these  with  the  differentials  of  sin~i  x,  cos'i  x,  tan-i  x,  cot-i  x. 


APPENDIX,  357 

10.    Following  the  method  by  which  relations  (5) -(9)  have  been  derived, 
show  that : 


fZf  hysin-i  ^\  =       ^^       ;  d[  hycos-i  ^  ^  =  ± 


hysin-i  ^  =  log  ^  +  ^^^^  +  ^^         hycos-i  ^  =  ±  i^  +  V^2  -  a'2 
a  a  a  a 

hytan-l  -  =  ^  log  ^^^t^  for  a:^  <  a' ;   hycot-i  -  =  |  log  ^^^^  for  x^  >  a^. 

11.    Assuming  the  relations  in  Ex.  10,  show  that  the  x-differentials  are  : 

x\  _  ,         dx       . 

fZfhytan-i^^  =  ^^^  fora:2<a2;    ^fhycot-i  ^\  =  _  _^L^  for  a:2;>«2. 
V  a/      a^-x^  \  a)         x^  —  d^ 

Compare  these  differentials  with  the  differentials  of  sin-i  -,  cos-i  -,  tan-^  -, 


12.  Assuming  the  relations  in  Ex.  10  as  definitions  of  the  inverse  hyper- 
bolic functions,  derive  the  definitions  of  the  corresponding  direct  hyperbolic 
functions.     (Suggestion.    Follow  the  plan  outlined  in  Ex.  8.) 

3.  Inverse  hyperbolic  functions  defined  as  integrals.  It  follows 
from  Ex.  11.  Art.  2,  that 

r_d^^  =  hysin-i-  +  c ;    f      ^^       =  ±  hycos-^  -  -f  c ; 

r--^,  =  ihytan-i-+c,(a)2<a2);    f-^  =  _  1  hycot "^  -  +  c, 
J  a"— or      a  a  J  x'—a  a  a 

Accordingly,  these  inverse  hyperbolic  functions  can  be  ex- 
pressed in  terms  of  certain  definite  integrals,  viz. : 


'^      dx,  1.^ ''^' +  V?/.^4-«^ 


Jo  V«2  +  a2  a  a' 

—J^^--  =  log  -^ =±  hycos-1  - ; 

a   Va;2-a2  a  a' 

jQa^-x2      2  a        a  —  u  a'  a' 

— i 7y  =  —  7r-^<^% =--hycot-i -,  i*2>a2. 


358  INFINITESIMAL   CALCULUS. 

These  relations  between  definite  integrals  and  inverse  hyperbolic 
functions  may  be  taken  as  definitions  of  the  functions. 

The  inverse  circular  functions  can  be  defined  by  integrals  which 
are  very  similar  to  the  integrals  appearing  in  the  definitions  of  the 
hyperbolic  functions.     Thus : 

dx              .     1  u               C^      d,x  -iU 

_  cin-l  I       =—  cos~ -. 


Va^  —  x^  ^  "^  "  Va^ 


f      '^^       ^  sin-''^.  ( 

Jo  y„2  _  3,2  a  J 

X 


""^     =itan-i».'  r^^^.  =  -ieot-'« 


\  o?  -\-  7?     a  a  Joo  a'-  +  x^         a 


EXAMPLES. 

1.  Find  the  area  of  the  sector  AOP  of  the  hyperbola  x^—y^  =  a^ 
(Fig.  106),  P  being  the  point  for  which  x  =  u.  Thence  show,  from  tlie 
definition  above,  that  hycos-i  -  is  the  ratio  of  twice  the  sector  AOP  to  the 
square  whose  side  is  a. 

2.  Find  the  area  of  the  sector  BOP'  bounded  by  the  ?/-axis,  the  arc 
BF  of  the  hyperbola  y^  —  x"^  =  cfi  (the  conjugate  of  the  hyperbola  in  Ex.  1), 
and  the  line  OP'  drawn  from  the  origin  to  the  point  P ,  P'  being  the  point  for 
which  X  =  u.  Then  show,  from  the  definition  above,  that  hysiir^  -  is 
the  ratio  of  twice  the  sector  BOP*  to  the  square  whose  side  is  a.  ^ 

3.  Sketch  the  curve  y(a^—x'^)=a^.  Calculate  the  area  between  this  curve, 
the  axes,  and  the  ordinate  for  which  x  =  u(u'^<::_a^) .  Show  that  hytanT^  -  is  the 
ratio  of  this  area  to  the  area  of  the  square  ivhose  side  is  a.  ^ 

4.  Sketch  the  curve  y(x'^  -  a"^)  =  a^.  Calculate  the  area  bounded  by  this 
curve,  the  x-axis,  and  the  ordinate  at  x  =  u(ifi'>a'^).  Show  that  hycot~^  -  is 
the  ratio  of  this  area  to  the  area  of  the  square  whose  side  is  a.  ^ 

4.   Geometrical  relations  and  definitions  of  the  hyperbolic  functions. 

In  Eig.  105  P  is  any  point  (x,  y)  on  a  circle  x^  +  ?/-  =  al  Let  the 
area  of  the  sector  AOP  be  denoted  by  u  and  the  angle  AOP  by  6. 
Then,  by  plane  trigonometry, 

2^  =  ia2^;   whence,  ^  =  ^-  (1) 

In  Fig.  106  P  is  any  point  on  a  rectangular  hyperbola  x^—y^—a^. 
(The  a  of  the  hyperbola  bears  no  relation  whatever  to  the  a  of 


APPENDIX. 


359 


the  circle.)      Let  the  area  of  the  sector  AOP  be  denoted  by  u. 
Then 

u  =  area  0PM  —  area  AP3I  =^xy  —  i    Va^  —  or  dx  j 


whence,  «  =  ?-  log  ^  +  ^^'-»'    =  ^  log  £±l.t 
2  a  2  a 


(2) 


O 

From  (2),       log^tJ  =  ^5    whence,  ^±^  =  e< 


X      0 

Fig.  105.  Fig.  106. 

From  equations  (3),  on  addition  and  subtraction, 

2m  " 
2m  2u  2u  _2m  -^ 


2m 
6*2  _^  g    a2 


(4) 


*  That  is,  M  =  I  a2  hycos-i  -  ;  whence,  hycos-i  ^  =  ?^. 
a  a      a^ 

t  If  a  =  1,  log  (x  +  2/)  =  2  M  =  twice  area  AOP.  On  account  of  the  relation 
between  natural  logarithms  {i.e.  logarithms  to  base  e)  and  the  areas  of  hyper- 
bolic sectors,  natural  logarithms  came  to  be  called  hyperbolic  logarithms. 
The  connection  between  these  logarithms  and  sectors  was  discovered  by 
Gregory  St.  Vincent  (1584-1667)  in  1647- 


360  INFINITESIMAL   CALCULUS. 

Relations  (4)  lead  to  geometrical  definitions  of  the  hyperbolic  func- 
tions. These  definitions  are  given  in  the  following  scheme.  This 
scheme,  supplemented  by  relation  (1),  also  shows  the  close  geo- 
metrical analogies  existing  between  the  hyperbolic  and  the  circular 
functions. 

N.B.  In  Figs.  105, 106  the  a  and  u  of  the  circle  are  not  related 
in  any  way  to  the  a  and  u  of  the  hyperbola. 

In  a  hyperbola  or  —  y^  —  a- 
(Fig.  106),  if  P  is  any  point 
{x,  y)  and  u  =  area  sector  AOP, 

then         ^  =  hysin^, 


In  a  circle  x^-^y^  =  a'^  (Fig. 
105),  if  P  is  any  point  {x,  y) 
and  u  =  area  sector  AOP, 

then         ^  =  sin  -^, 
a             a^ 

X            2u 

-  =  cos , 

a             a'' 

• 

whence, 

• 

^r  =  sin-^  =  cos-^  = 
a-              a             a 

=  tan-i^. 

X 

--hycos^, 
a  a2 

J^  =  hytau2f5 
ac  a2 ' 


whence. 


^  =  hysin-i^  =  hycos-i^ 

=  hytan-i  K 

•  a? 

These  results  may  be  expressed  in  words  : 

The  circular  functions  may  be  defined  by  means  of  the  relations 
connecting  a  point  (x,  y)  on  the  circle  x^ -\- y"- =  a?  and  a  certain  cor- 
responding circular  sector;  and  the  hyperbolic  functions  may  be 
defined  by  means  of  the  relations  connecting  a  point  (x,  y)  on  the 
rectangular  hyperbola,  x^  —  y^  =  a^  and  a  certain  corresponding  hyper- 
bolic sector. 

Each  of  the  inverse  circular  functions  may  be  expressed  as  the  ratio 
of  tivice  the  a,rea  of  a  certain  sector  of  a  circle  of  radius  a  to  the 
square  described  on  the  radius  of  the  circle,  and  each  of  the  inverse 
liyperbolic  functions  may  be  expressed  as  the  ratio  of  twice  the  area  of 
a  certain  sector  of  a  rectangular  hyperbola  of  semi-axis  a  to  the  square 
described  on  this  semi-axis. 

(For  a  more  general  notion  see  Ex.  3  following.) 
The  term  hyperbolic  arose  out  of  the  connection  of  these  func- 
tions with  the  hyperbola. 


APPENDIX.  361 


EXAMPLES. 

1.  Show  that  hysin-i  |  =  hycos-^  f  =  hytan"i  i.     Represent  each  of  these 
functions  geometrically.     Compute  hysin-i  |.     {_A7is.  1.099.] 

2.  Show  that  hysin-i  |  =  hycos-^  f  =  hytan-i  |.     Represent  each  of  these 
functions  geometrically.     Compute  hysin~i  |.     \_Ans.  .693.] 

3.  Show  that,  if  AP  (Fig.  105)  is  an  arc  of  an  ellipse  b'^x^  +  a^y^  =  a^b^, 
and  u  denote  the  area  of  the  elliptic  sector  AOP,  it  is  possible  to  write 

?  =  cos^,    y=:sin^. 
a  ah      b  ab 

Also  show  that,  if  ^P  (Fig.  106)  is  an  arc  of  a  hyperbola  ^  -  |^  =  1>  and 
u  denote  the  area  of  the  hyperbolic  sector  AOP^  then 

and  thence  show  that 


^^:=hycos2j^,  M  =  hysin2i£. 

a  ab  b  ab 

(Williamson,  Integral  Calculus^  Arts.  130,  130  a.) 


X^         y.i 

4.  Show  that  a  point  P(x,  y)  on  the  ellipse  -3  +  b  =  1  ^^  ^^-  ^  ^^7  ^® 
represented  as  (acos^,  6sin^),  and  show  that  ^(=:  eccentric  angle  of  P) 

=  (2  area  sector  ^ OP -4- a6). 

x^     V 
Show  that  a  point  P{x,  y)  on  the  hyperbola  -^  —  ^  =  1  in  Ex.  3  may  be 

represented    as   (a  hycos  v,    6  hysin  v),    and    show  that    v  =(2  area  sector 
AOP---ab). 

5.  The  Gudermannian.     Suppose  that 

sec  (f>  4-  tan  <^  =  hycos  v  +  h}' sin  v.  (1) 

From    (1)    and   the    identities    sec- </>  —  tan^  </>  =  1,    hycos^?;  — 
hysin^u  =  l,  it  follows  that 

sec  <^  =  hycos  v,     (2)         tan  <^  =  hysin  v.  (3) 

Since  [see  Art.  2,  Ex.  2  («)]  log  (hycos  v  +  hysin  v)  =  v,  relation 

(1)  may  be  written  1      /       i   ,  *     xn  /a\ 

^  ^       -^  V  =  log  (sec  4>  +  tan  ^) ;  (4) 

that  is,  by  trigonometry, 


V  =  log  tan  (I  + 1)  =  2.302585  log^o  tan  (^  +  |J.  (5) 


362  INFINITESIMAL   CALCULUS. 

When  any  one  of  the  relations  (l)-(5)  holds  between  two  numbers 
V  and  cfi,  (f>  is  said  to  be  the  Gudermannian  of  v.*  This  is  expressed 
by  this  notation :  .  ,  z^s 

^  «j>  z=  gd  V,  (6) 

In  accordance  with  the  usual  style  of  inverse  notation  each  of 
the  relations  (4),  (6),  (6)  is  expressed 

v  =  gd-^^,  (7) 

The  second  members  of  (4)  and  (5)  are  more  frequently  denoted 
by  the  symbol  X(<1>),  which  is  read  "  lambda  (^,"  than  by  gd'^  cfj. 

Geometrical  representation  of  X  (<^)  or  gdr^  <^.  If  at  F  (x,  y)  in 
Fig.  106,  x  =  a  sec  ^,  then  y  =  a  tan  <^,  since  x^  —  y'^  =  a^.  On  mak- 
ing this  substitution  for  x  and  y,  it  can  be  deduced  that 

area  sector  AOP  =^a^  log  (sec  <^  +  tan  <^).  (8) 

From  this, 

log  (sec  <f>  +  tan  <^),  i.e.  X  (cf>)    (or  gd-^  <^)  =  ^  •  sec^Qr  ^OP     ^^^ 

a 

From  (4),  (6),  (8),  ^  =  gd (^  "  ''''^  ^^^P)-  (10) 

If  the  area  of  sector  AOP  be  denoted  by  u,  relations  (9)  and 
(10)  may  be  expressed 

9d-'<f>  =  —,    cf>  =  gd  —  - 

a^  a^ 

To  construct  ari  angle  whose  radian  measure  is  <f>.  In  Fig.  106, 
about  0  as  a  centre  with  a  radius  a  describe  a  circle.  From  M 
draw  a  tangent  to  this  circle,  and  let  the  point  of  contact  be  at  P* 
in  the  first  quadrant ;  and  draw  OP'.  Now  0M=  OP  sec  MOP ; 
i.e.  x  =  a  sec  MOP.  But,  according  to  the  hypothesis  in  the  last 
paragraph,  x  =  a  sec  <^.     Hence,  an^Ie  MOP'  =  <|>. 

If  a  point  P(x,  y)  on  the  hyperbola  x^  —  y^  =  a?-  (see  Ex.  4,  Art.  4)  be 
denoted  as  (a  sec  0,  atan0),  0  is  the  angle  which  has  just  now  been  con- 
structed. 

*  This  name  was  given  by  the  great  English  mathematician  Arthur  Cayley 
(1821-1895)  "in  honour  of  the  German  mathematician  Gudermann  (1798- 
1852),  to  whom  the  introduction  of  the  hyperbolic  functions  into  modern 
analytical  practice  is  largely  due."     (Chrystal,  Algebra,  Vol.  II.,  page  288.) 


APPENDIX,  863 

EXAMPLES. 

1.  Derive  result  (8). 

2.  (a)  Show  that,  0  and  v  being  as  in  equations  (l)-(7), 

gdv  =  sec"i  (hycos  v)  =  tan-i  (hysin  v)  =  cos"i  (hysec  v)  =  sin-i  (hytan  v) 

=  cot~i  (hycosec v)  =  cosec-^  (hycot v) ;  hytan ^^  =  tan x <|>. 
(6)  Show  that  gd'^  <i>  =  hycos-^  (sec  0)  =hysin-i  (tan  (p) ;  gd  x=2ta.n-^  e==—  ^- 

3.  (a)  Show  that  tlie  derivative  of  \(^(p)(i.e.  gd-^<p)  is  sec0.  (6)  Show 
that  X(— 0)  =  — X(0).  [Suggestion.  Show  that  X(— 0)+ X(0)=  logl.] 
(c)  Sketch  the  graph  of  X(0). 

4.  Show  that  i  hysec  w  rfu  =  gd  u  ;    (  sec  n  du  =  (/d"i  ?<. 

Note.  References  for  collateral  reading  on  hyperbolic  functions.  Gib- 
son, Calculus,  §§  G6,  111,  110  ;  Lamb,  Calculus,  Arts.  19,  23,  40,  44,  72,  98, 
Exs.  2,  3  ;  F.  G.  Taylor,  Calculus,  Arts.  62-80,  439  ;  W.  B.  Smith,  Infinitesi- 
mal Analysis,  Vol.  I.,  Arts.  99-113;  McMahon  and  Snyder,  Diff.  CaL, 
pp.  320-325.  For  further  information  see  Chrystal,  Algebra  (ed.  1889), 
Vol.  II.,  Chap.  XXIX.,  §§  24-31  (pages  276-291)  ;  the  notes  on  pages  288, 
289  contain  interesting  information  about  the  liistory  and  literature  of  the 
subject.  Also  see  Hobson,  Treatise  on  Plane  Trigonometry,  Chap.  XVI. 
An  excellent  account  of  hyperbolic  functions,  starting  from  the  geometrical 
standpoint  and  showing  practical  applications,  is  given  in  McMahon,  Hyper- 
bolic Functions  (i.e.  Merriman  and  Woodward,  Higher  Mathematics,  Chap. 
IV.,  pages  107-168). 

NOTE    B. 

INTRINSIC   EQUATION    OF   A   CURVE. 

1.  The  intrinsic  equation  of  a  curve.  Usually  the  equation  of  a 
curve  involves  either  the  Cartesian  coordinates  x  and  y  or  the 
polar  coordinates  r  and  6.  In  some  cases  the  intrinsic  equation 
is  especially  useful.  In  the  intrinsic  equation  of  a  curve  the 
coordinates  chosen  for  any  point  P  are  (a)  the  distance  of  P  from 
a  chosen  fixed  point  on  the  curve,  this  distance  being  measured 
along  the  curve,  and  (6)  the  angle  made  by  the  tangent  at  P  with  a 
chosen  fixed  tangent  of  the  curve.  These  coordinates  are  denoted 
respectively  by  s  and  <^.  The  relation  connecting  them,  f{s,  <f>)=0 
say,  is  called  the  intrinsic  equation  of  the  curve.  The  term 
intrinsic  is  used  because  the  coordinates  s  and  <f)  are  independent 
of  all  points  or  lines  of  reference  other  than  the  points  and 
tangents  of  the  curve  itself. 


864 


INFlNITEStMA L    CALCUL tfS. 


EXAMPLES. 

1.  Derive  the  intrinsic   equation    of  a  straight  line.     Let  AB  he  any- 

straight  line.      Let    0  be   the  chosen 
I I I I      fixed  point,  and  P(s,  0)  be  any  pomt 

on  the  line.     It  is  required  to  find  the 
equation  which   is   satisfied  by   s  and  0. 

The  direction  of  the  line  at  P  is  the  same  as  the  direction  at  0 ;  hence  the 
intrinsic  equation  is  0  =  0. 

2.  Derive  the  intrinsic  equa-  P{s,<P)  /~~y^ 
tion  of  a  circle  of  radius  a. 
Take  (Fig.  107)  0  for  the  fixed 
point,  and  the  tangent  at  0  for 
the  tangent  of  reference.  Let 
P(s,  0)  be  any  point  on  the 
circle.  Then  s  =  arc  OP  and 
0  =  angle  TBP.  Now  arc  OP 
=  a  •  angle  0  ;  i.e.  s  =  acp. 


Fig.  107. 


2.   Derivation  of  the  intrinsic  equation  of  a  curve.     The  intrinsic 
equation   of   a   curve   is   usually   derived   from   its  equation  in 

Cartesian  coordinates  or  from  its 
equation  in  polar  coordinates.  The 
general  method  of  doing  this  will 
now  be  shown. 

Let  the  equation  of  the  curve  be 

f(x,y)=0.  (1) 

Take  Q  for  the  fixed  point,  and 
the  tangent  at  0  for  the  tangent  of 
reference.     Take  any  point  P  on  the 
curve ;   let  its  Cartesian  coordinates 
be  X,  y,  and  its  intrinsic  coordinates  be  s,  tf). 
Express  s  in  terms  of  x,  y;  suppose  that 

s=Mx,y).  (2) 

Also  express  <^  in  terms  of  x,  y ;  suppose  that 

<l>=Mx,y).  (3) 

The  elimination  of  x  and  y  between  equations  (1),  (2),  (3),  will 
give  the  required  equation  between  s  and  <;^. 


Fig.  108. 


APPENDIX.  365 

Similarly,  let  the  polar  coordinates  of  P  be  r  and  dj  and  let 
the  polar  equation  of  the  curve  be 

F{r,6)  =  0.  (4) 

Express  s  in  terms  of  r,  6 ;  suppose  that 

s  =  F,{r,6).  (5) 

Also  express  <^  in  terms  of  r,  d\  suppose  that 

4>  =  F,{:r,e).  (6) 

The  elimination  of  r  and  6  between  equations  (4),  (5),  (6),  will 
give  the  required  equation  between  s  and  <^. 

Note.  A  tangent  parallel  to  the  x-axis  is  usually  chosen  for  the  tangent 
of  reference. 

EXAMPLES. 

1.  Derive  the  intrinsic  equation  of  the  hypocycloid 

x3  +  2/3  =  ai  (1) 

Take  the  cusp  on  the  positive  part  of  the  x-axis  for  the  fixed  point,  and 
the  tangent  there  for  the  tangent  of  reference.  Then  at  any  point  P(x,  y) 
on  the  arc  in  the  first  quadrant 

tan0  =  -(i/^-x^),  (2) 

and  s  =  I  a^  («^  -  x^).  (3) 

From  (1)  and  (2),  sec2  0  =  tan2  ^  +  i  =  a^  --  x^.    . 

Substitution  for  x^  in  (3)  gives  2s  =  Za  sin^  0, 

2.  If  in  Ex.  1  the  chosen  fixed  point  O  be  at  a  distance  h  along  the 
curve  from  the  cusp  and  the  chosen  fixed  tangent  (not  necessarily  at  0) 
make  an  angle  a  with  the  tangent  at  the  cusp,  show  that  the  intrinsic 
equation  of  the  hypocycloid  is 

2  (s+  &)  =3asin2(0  +  a). 

3.  Find  the  intrinsic  equation  of  the  cardioid  r  =  a(l  —  cos  6). 

Let  the  polar  origin  be  chosen  for  the  fixed  point,  and  the  tangent  there 
be  chosen  for  the  tangent  of  reference.     Let  P(x,  y)  be  any  point  on  the 

cardioid.    Then         s  =  (Jyj r^  -{- (^Yde  =  iai I  -  cos -Y  (1) 


366  INFINITESIMAL    CALCULUS. 


Also,  (Art.  60),    0  =  ^  +  tan-i  ^i^=d  +  tan-^ f tan  '^]  =  ^e.  (2) 

dr  V        2 ,' 


On  substituting  in  (1)  the  value  of  6  from  (2), 

s  =  4  af  1  —  cos^V 


4.  If  in  Ex.  3  the  chosen  fixed  point  be  at  a  distance  b  from  the  polar 
origin  and  the  chosen  tangent  of  reference  make  an  angle  a  with  the  tan- 
gent at  the  polar  origin,  show  that  the  intrinsic  equation  of  the  cardioid  is 


5.    Derive  the   intrinsic  equation  of  each  of  the   following  curves,  the 
fixed  point  and  the  fixed  tangent  being  as  indicated  :    (1)    tlie   catenary 

X  X 

?/  =  -  (e«  +  e  «),  the  vertex  and  tangent  thereat ;  (2)  the  parabola  ?/2  =  4  ax, 

6 
the  vertex  and  tangent  thereat ;    (3)  the  parabola  r  =  a  sec^  -,  as  in  (2)  ; 

(4)  the  cycloid  x  =  rt(^  — sin^),   ?/ =  a(l  —  cos  ^),  with  reference  to  (a) 

the  origin  and  tangent  thereat,   (h)  the  vertex  and  tangent  thereat ;  (5)  the 

logarithmic  spiral  r  =  ce"^  ;  (6)  the  semi-cubical  parabola  3  ay'^  =  2  x^,  the 
origin    and   tangent    thereat ;    (7)    the   curve 


(8)    the  semi-cubical   parabola  y^  =  ax^ ;    (9)    the   tractrix  x  =  Vc-  —  y'^  4- 
clog^-i — C"  —  y  ^  ^jjg  point  (0,  c).      (For  an  account  of  the  tractrix  and 

y 

for  various  problems   which  reveal   its  properties,   see  the   text-books  of 
Williamson,  Byerly,  Lamb,  and  F.  G.  Taylor,  on  the  calculus.) 

[Answers :    Ex.  5.    (1)    s  =  a  tan  0,      (2)    s  =  a  tan  <p  sec  0  +  a  log  tan 

1^  +  -],     (3)    as  in   (2),     (4)   (a)    s  =  4  a(l  -  cos  0),     (b)   s  =  4  a  sin  0, 
\2      4/  ,  X 

(5)    s  =  c(e«*-l),     (6)   9s  =  4a(sec30  -  1),     (7)   s  =  alogtan  (? -f  J  J, 
(8)  27s=8a(sec3  0-l),    (9)  s  =  clogsec  0.]  ^^      ^^ 


3.   Radius  of  curvature  derived  from  the  intrinsic  equation.     The 

radius  of  curvature  at  a  point  on  a  curve  can  very  easily  be 
deduced  from  the  intrinsic  equation.  For,  according  to  Arts.  146, 
147,  the  radius  of  curvature  being  denoted  by  R, 

M  =  '^. 


APPENDIX.  367 


EXAMPLES. 


1.  In  Art.  2,  Ex.  5  (1),  J?  =  a  sec2  <p. 

2.  Find  the  radius  of  curvature  for  each  of  the  curves  in  Art.  2,  Ex.  1, 
Ex.3,  Ex.  5  (4),  (5),  (6),  (9). 

[Answers:  Ex.  1.    |  a  sin  2  0  ;  Ex.  3.  |asin^;  Ex.  5(4).  (a)  4  a  sin  0, 

o 

(6)  4  a  cos  0  ;  (5)  a  ce""^  ;  (6)  *  a  sec^  0  tan  0  ;  (9)  c  tan  0.] 

Note.     On   ^Ae   intrinsic  equation  of  a  curve,  see  Todhunter,  Integral 
Calculus,  Arts.  103-119;  Byerly,  Integral  Calculus,  Arts.  114-123. 


NOTE    C. 

EVALUATION    OF    INDETERMINATE    FORMS. 

x^  —  4 

1.    Indeterminate  forms.     When  x  =:  1,  the  value  of  the  fraction  ;r 

X  -2 

is  3  ;  when  x  =  1.5,  its  value  is  3.5  ;  w^hen  x  =  1.9,  its  value  is  3.9  ;  ichen 
x  =  2,  the  fraction  takes  the  form  -;  when  x  =  2.1,  the  value  of  the  frac- 
tion is  4.1  ;  when  a:  =  2.5,  its  value  is  4.5.  Thus  the  fraction  has  definite 
values  when  x  =  '--,  1,  ••.,  1.5,  •••,  1.9,  2.1,  •••,  2.5,  •••.  It  is  reasonable 
to  conclude  that  it  has  a  definite  value  when  x  =  2.    Put  x  =  2  -{■  h;  then  the 

fraction  becomes  ^    ^  ^)   ~    ,  i,e,  4  +  A.     The  limiting  value  of  this  when  h 

2  +  h  —  2                              3-2  —  4 
approaches  zero,  is  4.     Accordingly,  lima.i2 =  4.    Thus  the  true,  or 

0  ^~  ^ 

limiting,  value  of  the  form  -  which  appeared  above  is  4. 

0  ^ 

The  form  -  is  usually  called  an  indeterminate  form.  This  name  is  a  mis- 
nomer ;  for,  as  will  presently  appear,  the  value  of  such  a  form  may  be  deter- 
mined. A  better  name,  perhaps,  is  an  undetermined  form,  or  an  elusive  form, 
or  an  illusory  form.  The  evaluation  of  expressions  taking  this  form  can  be 
effected  in  various  ways.  Several  of  these  ways  are  shown  in  text-books  on 
algebra,  and  will  not  be  discussed  here  ;  *  this  Note  is  concerned  only  with 
the  evaluation  of  illusory  forms  by  means  of  the  calculus. 

Note  1.  The  only  applications  of  this  Note  in  the  preceding  part  of  this 
book  are  in  Art.  165. 

*  In  text-books  on  Algebra  the  form  0  ^  0  is  often  called  a  vanishing 
fraction;  for  its  evaluation,  see  (among  others)  Hall  and  Knight,  Algebra 
(4th  edition),  §§  271,  272. 

There  are  various  illusory  forms  besides  -;  viz.  ^,  1*,  0"^,  co",  cc  •  0, 

00  —  CO.    Their  evaluation  will  be  found  to  depend  upon  the  evaluation  of  0  -^0. 


368  INFINITESIMAL   CALCULUS, 

Note  2.     The  chief  methods  used  in  algebra  for  evaluating  expressions 
having  the  form  0-^0,  are  : 

(a)  By  removing  common  factors  from  the  numerator  and  denominator. 

Thus  ^^-^=  (a;-2)(a;  +  2)  ^  ^  _^  g  ;  this,  when  x  =:  2,  is  equal  to  4. 
ic  — 2  (x  -  2)  X  1 

{}))  By  expanding  in  series.     For  example,  sin  a;  -^  x  takes  the  form  0  -=-  0 
when  ic  =  0.     On  using  the  expansion  for  sin  x  given  in  Ex.  2,  Art.  178,  it  is 

easily  seen  that  lim^^io =  1. 

X 

Note  3.     Illusory  forms   have   frequently   appeared  in   this  book ;    for 
instances,  see  Art.  14  (Exs.  11-14),  Art.  22,  and  Chapter  lY. 

2.   Evaluation  of  expressions  taking  the  form  ^ »     Suppose  that  f{x) 

and  0(x)  are  continuous  functions  of  x,  and  that  /(«)  =  0  and  0(a)  =  0. 
Suppose  further  tiiat  /(x  +  ]i)  and  0(x  +  h)  can  be  expanded  by  Taylor's 
formula  in  the  neighbourhood  of  x  =  a.     Let  it  be  required  to  determine  the 

true  value  of  '-^  • 
0(a) 

Now,  the  value  of  ^  =  lim„^o  f ^^  ^  ^^ .  (1) 

[In  what  follows,  the  expansion  of  /(«  +  1i)  is  obtained  by  writing  the 
expansion  of  /(x  +  h)  and  then  substituting  a  for  x  therein.] 
By  Taylor's  theorem  [Art.  176,  formula  (9)], 

f{a  +  h)  ^  f{a)  +  hf'{a  +  d^h)  ^  /(a  +  M) 
0(a  +  h)     0(a)  +  h(p'{a  +  e^h)      <t>'{a  +  d2h)  ' 

(The  0's  are  each  less  than  1.) 

Hence,  by  (1),  the  value  of      Z^  =  ZW  . 
^  ^  ^  0(a)      0'(a) 

If  /'(a)  and  0'(a)  are  both  zero,  then 

0(«  +  /^)     0(a)+;i0'(a)+|->"(a-fM)     ^"(«  +  ^^'^) 


Hence,  by  (1),  the  value  of         J^-^  =  ^^^^f^ 

0(a)      0"(a) 


On  proceeding  in  this  way  it  can  be  shown,  by  means  of  Taylor's  theorem, 
that,  if,  for  x  =  a,  /(x)  and  0(x)  and  all  their  derivatives  up  to  and  including 


(2) 


APPENDIX. 

their  nth  derivatives,  are  zero,  while  /("+^)(a)  and  ^("^^^(a),  are  not  both 
"'•"•""'"        the  value  «f|M  =  /»;"(«). 

Result  (2)  may  also  be  expressed  thus  : 

<j)(ic)  <|>>"+i)(ac) 

EXAMPLES. 

1.  Evaluate  ^^-::ii  when  x  =  2.     (See  Art.  1.) 

x-2  ^  ^ 

Valuex-2  ^^^  =  value^i2  ^(^^'*)  =  valuex-2  — '  =  4. 
x-2  D{x-2)  1 

2.  Evaluate  {x  —  sin  x)  -=-  x^  when  x  =  0.     In  this  case, 

,.,      x-sinx       ,.^      1-cosx*     i;^      sinx*      ,.       •  cosx     1 

linia^ -. =  limx-=o — r--; —  =  Iiuix-^ =  Jinii^— — -  =  - . 

x^  3x2  6x  b        b 

Xote.  The  labour  of  evaluating  /(a)  -h  ^(a)  may  be  lightened  in  the 
following  cases  : 

(a)  If,  in  the  course  of  the  reduction  a  factor,  say  ypix),  appears  in  both 
the  numerator  and  the  denominator,  this  common  factor  may  be  cancelled. 
[Compare  Art.  1,  Note  2  (a).] 

(h)  If  at  any  stage  during  the  process  of  evaluation  a  factor,  say  ^(x), 
appears  only  in  the  numerator  or  only  in  the  denominator,  and  ^(a)  is  not 
zero,  the  value  of  \}/{a)  may  be  substituted  immediately  for  i/'(x).  This  will 
lessen  the  labour  in  the  succeeding  differentiations. 

3.  Evaluate    the    following:     (1)    ^'~^',    when    x  =  0;     (2)     !i^::i?, 

X  X 

whenx^O;   (3)   ?!lz:^  when  x=a  ;  (4)  ^'~^"',  whenx=0;  (5)  kz^^?^, 
when;^  =  0.  ^-«  ^^^^ 

4.  Find  the  following  : 

(1)  lim^  {x~^y^\nx  ^2)  lim^.^  (^  "  5)^  ^og  (3  -  x)  . 

X  sin  (x  —  2) 

/-QN  1,-rv,       e==  + e-* +  2C0SX  — 4  ,..  ,.^       tan  x  — sinx 

(3)  lim^^y)  — !^ ^^— ;         (4)  lim^^ : ; 

X*  X  -  sm  X 

(5)lim^  l-^««^ 


cos  X  sin2  X 

{Answers :  Exs.  3.    log  -,  1,  wa«-i,  2,  ^  ;  Exs.  4.    25,  -  9,  |,  3,  ^.] 
6 


*  Which  is  in  the  form  0  -r-  0. 


B70  INFINITESIMAL   CALCULUS. 

3.   Evaluation  of  expressions  taking  the  form  ~ .     Suppose  that /(a)  =  00 

and  0(a)  =  00  ,  and  let  it  be  required  to  determine  the  value  of  ^-^  • 

0(a) 
_1_       • 

Now  ^-^  ^t^.    The  latter  is  in  the  form  -•    Application  of  result 
<p{a)     _1_  0 

(2)  Art.  2,  gives 

0'(«) 


value  of  Z^  -     t^^(^):]'  =  f  value  of   .^^.V  .  ^W  .  a) 

0(«)  ~      /(«)       V  0(«)/     /'(«)  v^ 

From  (1),     value  of  -^  =  -^^^  ;  i.e.  lim^^a  ^^  =  lim^-.„  ^^^. 
^  ^'  0(a)      0'(a)'  ^-"0(x)  ^-"0'(x) 

Similarly,  if  -'  ^  -'  is  also  illusory,  its  value  can  be  shown  to  be  -^    ^    ^ ; 
0'(«)  0"(a) 

and  so  on.     It  thus  appears  that  the  methods  for  evaluating  the  illusory  forms 
in  Arts.  2  and  3  are  the  same. 

EXAMPLES. 

1.  Evaluate  -^  when  aj  =  oo  .     (See  Art.  8,  Note  2.) 

logx 

X  1 

limx=oo  , =  lim^iao  7  =  lim^^  a?  =  oo . 

log  X  1 

X 

2.  Evaluate  — ,    — ,    — ,  when  x  =  ao. 

gx        gat        gx 

3.  Find:  (1)  lim^.o^^;   (2)  linv.  ^^^  ;  (3)  lim^.  *?^lA^. 

^  cotx'   ^  ^        ^2  sec3x    ^  ^        =^2  tana; 

[Answers :  Exs.  2.   0,  0,  0  ;  Exs.  3.   0,  -  3,  f  ] 

4.  Evaluation  of  other  indeterminate  forms.     The  evaluation  of  these 
forms  can  be  made  to  depend  on  Arts.  2,  3. 

(a)  The  form  0  •  ao .     Suppose  that  0(a)  =0  and  i/'(a)  =  co  ,  and  let  the 
value  of  0  (a)  •  \p  (a)  be  required. 

Now  0(a)  •  i/'(a)  =  <t)(d)  -f ,  which  is  in  the  form  - ;  also  0(a)  •  ^p(a) 

=  i^(a)  -H ,  which  is  in  the  form  55 .     Thus  expressions  having  the  form 

0(a)  «^ 

0  •  00  can  be  transformed  into  expressions  having  the  form  0  -^  0  or  00  -:-  00 . 


APPENDIX.  371 

EXAMPLES. 

1 


sec2  X 

m 


2-2x^     1 

2x  2' 


1.  Limx=o(a;  •  cotx)  =  linxcio -^—  ( i.e.  -  j  =  linia^  — 

tanicV       0/  sec' 

2.  Determine  :       (1)   lim^^  ( -  —  x\  tan  x  ;  (2)   Wvax^oo  a""  •  sin 
(3)  lim^^i  (x-\)  tan  ^.     { Answers ;  1,  m,  -  -•] 

(6)  The  form  00  —  00.  By  combining  terms  and  simplifying,  an  expres- 
sion having  the  form  oo  —  oo  may  be  reduced  to  a  definite  value,  or  to  one  of 
the  preceding  illusory  forms. 

EXAMPLES. 

3.  Lim^sf-^  -  -^-^  =  lim,^2  ^  ^,  ~  f  =  lim.^, 

V^^  -  4      X  -  2/  x2  — 4 

4.  Find:  lim,=i  f  ^^  - -J_V  Hm,^  jl  -  1  log  (1  +  x)  | , 

V^  —  1      log  x)  i.  a-     x2  J 

lim^,^ (x  —  Vx'^  —  oP').     [Answers:  ^,^,0.] 

(c)  The  forms  1*,  ooO,  00.  Suppose  that  [0(a)]'''(«)  is  in  one  of  these 
forms.     Put  u  =  [0(a)]'^(«) ;  then  log  u  =  \f/(a)  •  log  [0(a)]. 

The  second  member  is  in  the  form  0  •  00  ;  and,  accordingly,  log  u  may  be 
determined.     Then  the  value  of  u  can  be  deduced  from  the  value  of  log  u. 

EXAMPLES. 
1 

5.  Evaluate  (1  -  xy  when  x  =  0.     (The  form  then  is  1*.) 

Put  u=(l-xy]    then  log  u  =  ^og  (^  -^). 

x 

Accordingly,  Iima;io log u  =  limjj^of  ^^ — 1  =  —  1.     .-.  u  =-  when  x  =  0. 

\l-xj  e 

6.  Findlim;,io(a:^).     (This  form  is  O''.) 

Put  u  =  x'' ;    then  log  u  =  x  log  x. 

Accordingly,  1 

lim^c^ log u  =  lim^^ ^^^  =  limx^o  _    .^  =  limx=o {-  x)=0; 

consequently,  u  =  e^  =  1  when  x  =  0. 

7.  Evaluate  the  following:  (1)  [1  +  -]  when  x  =  00 ;  (2)  sinxta°* 
when  X  —  0  ;  (3)  x-^  when  x  =  oo  ;  (4)  (1  —  xy  when  x  =  00  ;  (5)  M  +  -  J 

/         l\x  _L  ^ 

when  X  =  00  ;    (6)  I  1  +  —  J     when  x  =  go  ;    (7)  x*-i  when  x  =  1  ;    (8)  x*-i 

when  X  =  CO  ;  (9)  x^^n^  when  x  =  0.     [^?isiocrs;  (1)  e,   (2)  1,   (.3)  1,   (4)  1, 
(5)  00,  (6)  1,  (7)  e,  (8)  1,  (9)  1.] 


372  INFINITESIMAL   CALCULUS. 

Note.  References  for  collateral  reading  on  illusory  forms.  For  a 
fuller  discussion  on  the  evaluation  of  expressions  in  these  forms,  and  for 
many  examples,  see  McMahon  and  Snyder,  Diff.  Cal.,  Chap.  V.,  pages  115- 
131  ;  F.  G.  Taylor,  Calculus,  Chap.  XII.,  pages  136-148  ;  Echols,  Calculus, 
Chap.  VII.  ;  also  Gibson,  Calculus,  Arts.  161,  162.  For  a  general  treatment 
of  the  subject  see  Chrystal,  Algebra,  Vol.  II.,  Chap.  XXV. 

NOTE    D. 

APPLICATIONS   TO   MECHANICS. 

N.B.  For  a  full  explanation  and  discussion  of  the  mechanical  terms  in 
this  note,  see  text-books  on  Mechanics. 

1,  Mass,  density,  centre  of  mass.  For  this  note  the  following 
definition  of  mass  may  serve :  The  mass  of  a  body  is  the  quayitity 
of  matter  which  the  body  contains.*  The  principal  standards  of 
mass  are  two  particular  platinum  bars;  the  one  is  the  "imperial 
standard  pound  avoirdupois,"  which  is  kept  in  London,  and  the 
other  is  the  ''  kilogramme  des  archives,"  which  is  kept  in  Paris. 

Note.  The  weight  of  a  body  is  the  measure  of  the  earth's  attraction  upon 
the  body,  and  depends  both  on  the  mass  of  the  body  and  its  distance  from 
the  centre  of  the  earth.  The  same  body,  while  its  mass  remains  constant, 
has  different  weights  according  to  the  different  positions  it  takes  with  respect 
to  the  centre  of  the  earth. 

The  density  of  a  body  is  the  ratio  of  the  measure  of  its  mass  to  the  measure 
of  its  volume  ;  that  is,  the  density  is  the  number  of  units  of  mass  in  a  unit  of 
volume.  The  density  at  a  point  is  the  limiting  value  of  the  ratio  of  (the 
measure  of)  the  mass  of  an  infinitesimal  volume  about  the  point  to  (the 
measure  of)  the  infinitesimal  volume.  A  body  is  said  to  be  homogeneous  when 
the  density  is  the  same  at  all  points.  If  a  body  is  not  homogeneous,  the  "den- 
sity of  a  body,"  defined  above,  is  the  average  or  mean  density  of  the  body. 

Centre  of  mass.  Suppose  there  are  particles  whose  masses  are  wii, 
?>i2,  mg,  ••♦,  and  whose  distances  from  any  plane  are,  respectively, 
dj,  c?2,  cfg,  •••.     Let  a  number  D  be  calculated  such  that 

J)  ^  midiH-mg^^a  +  mgC^g"-.     ^.  ^    ^^^  ^  ^  S  md 

Wi  -|-  m.2  +  mg  +  •  •  •    '  2»i 

For  any  given  plane,  D  evidently  has  a  definite  value. 

*  A  real  definition  of  mass,  one  that  is  strictly  logical  and  fully  satisfac- 
tory, is  explained  in  good  text-books  on  dynamics  and  mechanics.  (For 
example,  see  MacGregor,  Kinematics  and  Mechanics,  2d  ed..  Art.  289.) 


APPENDIX.  373 

If  (oTi,  ?/i,  Zy),  (x.2,  2/2,  2^2),  (a^3,  Vs,  ^s),  ••,  respectively,  be  the  coordi- 
nates of  these  particles  with  respect  to  three  coordinate  planes  at 
right  angles  to  one  anothei",  then  the  point  (x,  y,  z),  such  that 

»  =  -^— J    y=^r^^    2;=——,  (1) 

2m  2m  2im 

is  called  the  centre  of  mass  of  the  set  of  particles. 

If  the  matter  "  be  distributed  continuously  "  (as  along  a  line, 

straight  or  curved,  or  over  a  surface,  or  throughout  a  volume),  and 

if  Am  denote  the  element  of  mass  about  any  point  (a;,  y,  z),  then, 

on  taking  all  the  points  into  consideration,  equations  (1)  may  be 

written : 

X  =  l™A>»^S.t--Am^  and  similarly  for  y  and  z.  (2) 

hm^,^oSAm 

From  (2),  by  the  definition  of  an  integral, 

(  oc  din  \  y  dm  \  z  dm 

^  =  ^ ,V  =  ^ ,   S  =  =L (3) 

I  dm  I  dm  I  dm. 

If  p  denote  density  of  an  infinitesimal  dv  about  a  point,  then 

dTU  =  pdv  (4) ;    and,  on  integration,  m=\p dv,  (5) 

Ex.  Write  formulas  (3),  putting  pdt^  for  rim. 

Suppose  that  the  body  is  not  homogeneous;  that  is,  suppose 
that  the  density  of  the  body  varies  from  point  to  point.  Let  p 
denote  the  density  at  any  point  (x,  y,  z),  let  dv  denote  an  infini- 
tesimal volume  about  that  point,  and  let  p  denote  the  average  or 
mean  density  of  the  body.     Then 


mass  of  body  _)P^^ 
^  ~  vol.  of  body  ~    C^^  ' 


Note.  The  term  centre  of  mass  is  used  also  in  cases  in  which  matter  is 
supposed  to  be  concentrated  along  a  line  or  curve,  or  on  a  surface.  In  these 
cases  the  terms  line-density  and  surface-density  are  used. 


374 


INFINITESIMAL   CALCUL US. 


EXAMPLES. 

1.   In  a  quadrant  of  a  thin  elliptical  plate  whose  semi-axes  are  a  and  6, 
the  density  varies  from  point  to  point  as  the  product  of  the  distances  of  each 

point  from  the  axes.  Find  the  mass, 
the  mean  density,  and  the  position  of 
the  centre  of  mass,  of  the  quadrant. 
Choose  rectangular  axes  as  in  the  figure. 
At  any  point  P(x,  y),  let  p  denote  the 
density  and  dm  denote  the  mass  of  a 
rectangular  bit  of  the  plate,  say,  dx  •  dy. 
Let  M  denote  the  mass,  p  the  mean 
density,  and  (x,  y)  the  centre  of  mass, 
of  the  quadrant. 
Now  dm  =  p  dx  dy. 


Fig.  109. 


But  pccxy  ;  i.  e. 


p  =  kxy,  in  which  k  denotes  some  constant. 


Accordingly,  M=\dm=\        \     «  kxy  dydx  —  \k  a^h"^ 

_  _    mass  of  quadr; 
volume  of  quadi 


mass  of  quadrant    _\k  a^lP- 
volume  of  quadrant       \  irab 


kab 

27r  ' 


•V^^372 


dy  dx 


Here 


Similarly,         y 


^^kam 
1  ka^h-^ 


ha. 


dv  31 

Hence,  centre  of  mass  is  (y^^  a,  j^  b). 


2.  Find  the  centre  of  mass  of  a  solid 
hemisphere,  radius  a,  in  which  the  density 
varies  as  the  distance  from  the  diametral 
plane.     Also  find  the  mean  density. 

Symmetry  shows  that  the  centre  of  mass 
is  in  OY. 

Take  a  section  parallel  to  the  diametral 
plane  and  at  a  distance  y  from  it. 

The  area  of  this  section 

=  TT  .   CP^  =  7r(a2  _  J/2). 

For  this  section,  p  coy,  i.e.  p  =  ky,  say. 


:r 

r^"       ^ 

/^^ ' 

-;/  \ 

/ 

/ 

■y^~-.\ 

^^^^ 

Fig.  110. 


Then 


Also 


Jo' 


w(a^-y^)dy     kir  \   y'i(a^  -  y^)dy 


r 


X' 


p7r(a2  _  y2)dy 


■|;Ka2 


y^)dy 


vol.        I  7ra3       ^ 


This  is  the  density  at  a  distance  |  a  from  the  diametral  plane. 


APPENDIX. 


375 


3.  The  quadrant  of  a  circle  of  radius  a  revolves  about  the  tangent 
at  one  extremity.  Find  the  position  of  the  mass-centre  of  the  surface 
thus  generated.  In  this  case  let  the 
"surface-density"  be  denoted  by  p. 
Symmetry  shows  that  the  mass-centre 
is  in  the  line  PL.  Let  y  denote  the 
distance  of  the  mass-centre  from  OX. 

In  PL  take  any  point  iV,  at  a  dis- 
tance ?/,  say,  from  OX.  Through  N 
pass  a  plane  at  right  angles  to  PZ, 
and  pass  another  parallel  plane  at  an 
infinitesimal  distance  dy  from  the  first 
plane.  These  planes  intercept  an  infini- 
tesimal zone,  of  breadth  ds  say,  on  the 
surface  generated. 

Area  of  this  zone  =  2  tt  •  CiY-  c?s  =  2  7r(J/iY—  MC)ds. 


Now,  at  C  (x,  y) 


x^  +  y^  =  a^. 


Accordingly,     ds  =  Jl  ^  I^^Y  ■  dy  =  -^ 
Hence,  area  zone  =  2ir(a  —  Va'^  —  y'^) 


-dy. 


Va^- 


—  dy  =  2Tra( ■ 

2/2  VV^ 


.:y  = 


{"^py  •  (2  TT  .  CN-ds)      2  TT  ap^^y 


V  «2  -  y2 


-\yy. 

-l\ly 


p  -  area  zone 


2.ap^J 


-2 


876  a. 


iVfy 


4,  In  the  following  lines,  curves,  surfaces,  and  solids,  find  the  posi- 
tion of  the  centre  of  mass ;  and,  in  cases  in  which  the  matter  is  not  dis- 
tributed homogeneously,  also  find  the  total  mass  and  the  mean  density 
("line-density,"  "surface-density,"  or  "density,"  as  the  case  may  be). 
(The  density  is  unitorm,  unless  otherwise  specified.) 

(1)  A  straight  line  of  length  I  in  which  the  line-density  varies  as  (is  k 
times),  (a)  the  distance  from  one  end ;  (6)  the  square  of  this  distance ;  (c)  the 
square  root  of  this  distance. 

(2)  An  arc  of  a  circle,  radius  r,  subtending  an  angle  2  a  at  the  centre. 

(3)  A  fine  uniform  wire  forming  three  sides  of  a  square  of  side  a. 

(4)  A  quadrantal  arc  of  the  four-cusped  hypocycloid. 

(5)  A  plane  quadrant  of  an  ellipse,  semi-axes  a  and  b. 


376  INFINITESIMAL   CALCULUS. 

(6)  The  area  bounded  by  a  semicircle  of  radius  r  and  its  diameter, 
(a)  when  the  surface  density  is  uniform  ;  (&)  when  the  surface  density  at 
any  point  varies  as  (is  k  times)  its  distance  from  the  diameter. 

(7)  The  area  bounded  by  the  parabola  \/x-{-  Vy  =  Va  and  the  axes. 

(8)  The  cardioid  r  =  2  a(l  -  cosd). 

(9)  A  circular  sector  having  radius  r  and  angle  2  a. 

(10)  The  segment  bounded  by  the  arc  of  the  sector  in  Ex.  (9)  and  its  chord. 

(11)  The  crescent  or  lune  bounded  by  two  circles  which  touch  each  other 
internally,  their  diameters  being  d  and  ^d,  respectively. 

(12)  The  curved  surface  of  a  right  circular  cone  of  height  h^  (a)  when 
the  surface  density  at  a  point  varies  as  its  distance  from  a  plane  which  passes 
through  the  vertex  and  is  at  right  angles  to  the  axis  of  the  cone ;  (b)  when 
the  surface  density  is  uniform. 

(13)  A  thin  hemispherical  shell  of  radius  a,  in  which  the  surface  density 
varies  as  the  distance  from  the  plane  of  the  rim. 

(14)  A  right  circular  cone  of  height  h  in  which,  (a)  the  density  of  each 
infinitely  thin  cross-section  varies  as  its  distance  from  the  vertex;  (&)  the 
density  is  uniform. 

(15)  Show  that  the  mass-centre  of  a  solid  paraboloid  generated  by  revolving 
a  parabola  about  its  axis,  is  on  the  axis  of  revolution  at  a  point  two-thirds  the 
distance  of  the  base  from  the  vertex. 

(16)  A  solid  hemisphere  of  radius  r,  (a)  when  the  density  is  uniform  ; 
(&)  when  the  density  varies  as  the  distance  from  the  centre. 

(17)  Show  that  the  mass-centre  of  the  solid  generated  by  the  revolution 
of  the  cardioid  in  Ex.  (8)  about  its  axis,  is  on  this  axis  at  a  distance  |  a  from 
the  cusp. 

(18)  If  the  density  /o  at  a  distance  r  from  the  centre  of  the  earth  is  given 

by  the  formula  p  =  po  5ilLJ.  ^  in  which  po  and  k  are  constants,  show  that  the 

earth's  mean  density  is  „:     7  n      ?  td        7  r> 

•^  Q      sm  kE  —  kR  cos  kR 

^  k^R' 

in  which  R  denotes  the  earth's  radius.     (Lamb's  Calculus.) 

[Answers :  (1)  |  ?  from  that  end,  M=  I  kl^,  ~p  =  \kl;  (h)ll,  M-^  kl\ 
p  =  ^kl'^ ;  (c)  ^  I,  M  =  I  kl'2,  p  =^kli.  (2)  On  radius  bisecting  the  arc  at  dis- 
tance r from  centre.     (3)  At  a  distance  4  a  from  the  centre  of  the 

a  ^  ^  "  . 

square.     (4)  Point  distant  |  a  from  each  axis.     (5)  Point  distant  —  from 

46  ^^      4a 

axis  2  a,  —  from  axis  2  h.     (6)  (a)  On  middle  radius,  at  point  distant  — 

3  TT  3  TT 

from  the  diameter  ;  (h)  On  middle  radius,  at  point  .580  a  from  the  diameter, 
mean  density  =  .4244  maximum  density.  (7)  Point  distant  I  a  from  each 
axis.     (8)  The  point  (tt,  |  a).     (9)  On  middle  radius  of  sector,  at  distance 

f  r from  the  centre.     (10)  On  the  bisector  of  the  chord,  at  distance 


APPENDIX.  '  377 


^^"  ^^  from  the  centre.     (11)  On  the  diameter  through  the  point 


a  —  sin  a  cos 

of  contact  and  distant  |^  d  from  that  point.  (12)  (a)  On  the  axis,  at  distance 
I  h  from  the  vertex  ;  (&)  on  axis,  at  distance  |  h  from  vertex.  (13)  On  the 
radius  perpendicular  to  the  plane  of  the  rim,  at  a  distance  |  a  from  the  centre. 
(14)  («)  On  the  axis,  ^h  from  the  vertex;  the  mean  density  is  the  same  as 
the  density  at  the  cross-section  distant  f  h  from  the  vertex  ;  {h)  on  the  axis, 
at  a  distance  f  h  from  the  vertex.  (16)  (a)  On  a  radius  perpendicular  to  the 
base,  at  a  distance  .375  r  from  it;  (6)  on  radius  as  in  (a),  at  distance  Ar 
from  the  base.] 

2.  Moment  of  inertia.  Radius  of  gyration.  These  quantities  are 
of  immense  importance  in  mechanics  and  its  practical  applications. 

Moment  of  inertia.  Let  there  be  a  set  of  particles  whose  masses 
are,  respectively,  m^,  m^,  ?%,  •••,  and  whose  distances  from  a  chosen 
fixed  line  are,  respectively,  )\,  r^,  rg,  •••.     The  quantity 

mi/'i^  +  ^»'2^'2"  +  »i3>*3'  +  •••)  *-6.  5 ^rir^  (1) 

is -called  the  moment  of  inertia  of  the  set  of  particles  with  respect 
to  the  fixed  line,  or  axis,  as  it  is  often  called.  It  is  evident  that 
for  any  chosen  line  and  system  of  particles  the  moment  of  inertia 
has  a  definite  value.  In  what  follows,  the  moment  of  inertia  will 
be  denoted  by  J. 

It  can  be  shown,  by  the  same  reasoning  as  in  Art.  (1),  that 
definition  (1)  can  be  extended  to  the  case  of  any  continuous  dis- 
tribution of  matter  (whether  along  a  line  or  curve,  or  over  a  sur- 
face, or  throughout  a  solid)  and  any  chosen  axis;  thus. 


=  \  r2  dm, 


in  which  r  denotes  the  distance  of  any  point  from  the  axis,  and 
dm  an  infinitesimal  element  of  mass  about  that  point. 

Radius  of  gyration.     In  the  case  of  any  distribution  of  matter 
and  a  fixed  line,  or  axis,  the  number  k,  which  is  such  that 

..      the  n^oment  of  inertia      .f>--^'»^ 
the  mass  f^.,^  ' 

is  called  the  (length  of  the)  radius  of  gyration  about  that  axis. 


378  • 


INFINITESIMAL    CALCULUS. 


EXAMPLES. 

1.    Find  the  radius  of  gyration  about  its  line  of  symmetry  of  an  isosceles 
triangle  of  base  2  a  and  altitude  h. 

The  density  per  unit  of  area  will  be  denoted  by  p. 


Fig.  112. 


Let  P  be  any  point  in  the  triangle,  and  make  the  construction  shown  in 
the  figure.     Denote  NO  by  y. 


Then     A;^  =:  ^  2  PN^  -p-dx  dy  over  AOC 
2p  ■  dx  dy  over  ABC 


ry=n  p=z.v^2 


dxdy 


p  ah 


Now 


LN_ 
AO 


CN    .^   LN 
CO'    '  '     a 


h  —  V 


whence  LN=  -(h  —  y). 
h 


/.  k^ 


i^  =  la2.  whence  /fcz:.^. 
ah        6  y/Q 


In  this  example,  the  moment  of  inertia  is  ^  a%. 

2.  Show  that  the  moment  of  inertia  of  a  homogeneous  thin  circular  plate 
about  an  axis  through  its  centre  and  perpendicular  to  its  plane  is  ^  pir  a*,  in 
which  p  denotes  the  surface  density,  and  that  its  radius  of  gyration  is  |  aV2. 

I  On  using  polar  coordinates,  I  =  i  r^ .  (^^,^  —  i  r-  •  p  •  dA  =  p\      I   r^-rdrdd.  \ 


3.  Find  the  moment  of  inertia  of  a  solid 
homogeneous  sphere  of  radius  a  about  a 
diameter,  m  being  the  mass  per  unit  of 
volume.  Suppose  that  the  sphere  is  gener- 
ated by  the  revolution  of  the  semicircle  APB 
about  the  diameter  AB.  Let  rectangular 
axes  be  chosen  as  in  the  figure.  At  any 
point  P(x,  y)  on  the  semicircle  take  a  thin 
rectangular  strip  PN  at  right  angles  to  AB 


APPENDIX.  379 

and  having  a  width  Aa;.     This  strip,  on  the  revolution,  generates  a  thin  circu- 
lar plate.     It  follows  from  Ex.  2,  since  m  is  the  mass  per  unit  of  volume,  that 

/  of  this  plate  about  AB  =  -  tt  •  PN^  •  ^x. 

.'.  I  of  sphere  =  2  -tt  •  PN^Ax  from  A  to  B 
2 

=  2  .  ^  P  (a2  -  x'^ydx  =  j%  mira^ 
2  Jq 

Here,  on  denotin^the  mass  of  the  sphere  by  M, 
M  =  ^  mira^  ; 
hence,  ^  =  f  ^«^  > 

accordingly,  A:^  =  |  a2  . 

and  thus,  k  =  .632  a. 

4.  Find  the  moment  of  inertia  and  the  square  of  the  radius  of  gyration 
in  each  of  the  following  cases  : 

(Unless  otherwise  specified,  the  density  in  each  case  is  uniform.  The 
mass  per  unit  of  length,  surface,  or  volume  is  denoted  by  nij  and  the  total 
mass  by  M.) 

(1)  A  thin  straight  rod  of  length  ?,  about  an  axis  perpendicular  to  its 
length  :  (a)  through  one  end  point,  (&)  through  its  middle  point. 

(2)  A  fine  circular  wire  of  radius  a,  about  a  diameter. 

(3)  A  rectangle  whose  sides  are  2  a,  2  6:  (a)  about  the  side  2  6, 
(&)  about  a  line  bisecting  the  rectangle  and  parallel  to  the  side  2  b. 

(4)  A  circular  disc  of  radius  a  :  (a)  about  a  diameter,  (6)  about  an 
axis  through  a  point  on  the  circumference,  perpendicular  to  the  plane  of 
the  disc,  (c)  about  a  tangent. 

(5)  An  ellipse  whose  semi-axes  are  a  and  b  :  (a)  about  the  major  axis, 
(&)  about  the  minor  axis,  (c)  about  the  line  through  the  centre  at  right 
angles  to  the  plane  of  the  ellipse. 

(6)  A  semicircular  area  of  radius  a,  about  the  diameter,  the  density 
varying  as  the  distance  from  the  diameter. 

(7)  A  semicircular  area  of  radius  a,  about  an  axis  through  its  centre  of 
mass,  perpendicular  to  its  plane. 

(8)  A  rectangular  parallelopiped,  sides  2  a,  2  6,  2  c,  about  an  edge  2  c. 

(9)  A  right  circular  cone  (height  =  h,  radius  of  base  =:  6),  about  its  axis. 

(10)  A  thin  spherical  shell  of  radius  a,  about  a  diameter. 

(11)  A  sphere  of  radius  a,  about  a  tangent  line. 

(12)  A  right  circular  cylinder  (length  =  I,  radius  =  R)  :  (a)  about  its 
axis,  (6)  about  a  diameter  of  one  end. 


380  INFINITESIMAL    CALCULUS. 

(13)  A  circular  arc  of  radius  a  and  angle  2  a  :  (a)  about  the  middle 
radius,  (&)  about  an  axis  through  the  centre  of  mass,  perpendicular  to  the 
plane  of  the  arc,  (c)  about  an  axis  through  the  middle  point  of  the  arc, 
perpendicular  to  the  plane  of  the  arc  [Lamb's  Calculus,  Exs.,  ;j»i;XXIX.]. 

[Answers:    (1)   (a)   lml\  ^l';    (b)  i^ml^,    ^^P.       (2)   \Ma'^,   -i  a^. 

(3)  (a)  r-  =  ^,ar-,  (h)  Tc^  =  \a'^.     (4)   (a)  k^  =  \a^-  (b)  k^  =  ^a^;  (c)  k^ 
=  fa2.       (o)   (a)  \Mb-^;    (b)  \Ma'-,    (c)  \  M{a:^ -V  If-) .      (6)  |  Jifa^,  |a-. 

(7)  k^  =  [\-  ^\  a\     (8)  A:2  =  K«'  +  &'-^) •     (9)  ^V  ^'^'t  ft^/i,  ^\  b'^     ( 10)  ^'^ 

=  |a2.       (11)    A.-^^laS.       (12)    («)   I=\MB^;    (b)    /=  iV/(i  i?^  +  i  Z2). 

(13)    (a)    ^•2  =  i«-2(l-«"y-^^);       (/,)    /.■3  =  «2(i_«mia^.       (e)    A:^  = 

2a2fl-^'"-^    ' 


J 


QUESTIONS    AND    EXERCISES    FOR   PRACTICE 
AND   REVIEW. 


3j<K< 


A  large  number  of  examples  are  contained  in  several  works  on 
calculus,  in  particular  in  those  of  Todhunter,  Williamson,  Lamb, 
Gibson,  F.  G.  Taylor,  and  Echols.  Special  mention  may  also  be 
made  of  Byerly's  Problems  in  Differential  Calculus  (Ginn  &  Co.). 
Exercises  of  a  practical  and  technical  character,  which  are  con- 
cerned with  mechanics,  electricity,  physics,  and  chemistry,  will 
be  found  in  Perry,  Calculus  for  Engineers  (E.  Arnold)  ;  Young 
and  Linebarger,  Elements  of  the  Differential  and  Integral  Calculus 
(D.  Appleton  &  Co.)  ;  Mellor,  Higher  Mathematics  for  Students  of 
Chemistry  and  Physics  (Longmans,  Green  &  Co.).  Many  of  the 
following  examples  have  been  taken  from  the  examination  papers 
of  various  colleges  and  universities. 

CHAPTERS   II.,  III.,  IV. 

1.  Explain  what  is  meant  by  a  continuous  function. 

2.  Explain  what  is  meant  by  a  discontinuous  function.     Give  examples. 

3.  (1)  Given  that /(a;)  =  ic2  _|_  2  and  F{x)=4:+Vx,  calculate /{^(a:)} 
and  F{f(x)}.     (2)  If  /(x)  =  ^^^,  show  that    /(^)-/(y)   =  -^.nl-.     (3)  If 

y=f(x)  =  —^ — -   and   z—f{y),    calculate   ^r   as   a  function  of  x.     (4)  If 

4  —  7  X 

0  y 1 

y  =  0(x)  = ,  show  that  x  =  </)(y),  and  show  that  x  =  (p^(x),  in  which 

^x  —  2  T  -4-  1 

02(x)  is  used  to  denote  <p{<p(x)},  not  {0(x)}2.      (5)   If  f{x)='^^-^—^,  show 

that  /2(x)  =  X,  /3(x)  =/(x),  /4(x)  =  x.     (6)  If  v  =  f(x)  =  ^^±-^,  show  that 

ex  —  a 

^  =  f(y)-     (")  If  /(a^»  y)  =  «^^  +  bxy  +  c?/2,  write  /(?/,  x) ,  /(x,  x) ,  and  f(y,  y). 

4.  Define  the  differential  coefficient  of  a  function  of  x  with  regard  to  x. 
State  what  is  the  interpretation  of  the  differential  coefficient  being  positive 
or  negative. 

381 


382  INFINITESIMAL    CALCULUS. 

5.  Give  a  geometrical  interpretation  of  ^  ^  when  x  and  y  are  connected 
by  the  relation  /(x,  y)=Q  oi  y  =  (t>(x).  ^ 

6..  Show  that  the  derivative  of  a  function  with  respect  to  the  variable 
measures  the  rate  of  increase  of  the  function  as  compared  with  the  rate  of 
increase  of  the  variable. 

7.  Find  geometrically  the  differential  coefficients  of  cos  x  and  sin  x. 

8.  Deduce  from  first  principles  the  first  derivatives  of  x'*,  sinx,  tanx, 

X 

tan-i  aj,  log„a;,  a%  a^os*:,  log  sin  — 

u 

9.  Find  the  derivatives  of  —  and  uv^  with  respect  to  a;,  where  u  and  v 

are  functions  t)f  x. 

10.  Investigate  a  method  of  finding  the  derivative  with  respect  to  cc  of  a 
function  of  the  form  {/(a;)}'/»(^\  and  apply  it  to  differentiate  x^i+^^. 

11.  Differentiate        ^'"      ,  ^^gJ^^),  e-cos'-mx,  xe-^,  log^±«-^^, 

{\+x^y  X  a  +  6cosx 

gtan-i x^   tan-1  e==,  x™e"^ sin** x,  log  /2x_^HLi2n^ 

a;2  —  1 


12.    Show  that  (1)  Dsin-iJ^^ ^  =  Z>cos-1a/^^^ — -;   (2)  i>sin-i^^ 


6x 

+  I)sin-.^(''-^-»')(l-^')=0. 

a  +  &x 

13.  If  x^z/S  4-  cos  X  —  sin  X  tan  ?/  —  sin  ?/  =  0,  show  that 

dy  _  (  —  2  x?/^  +  sin  x)  cos^  y  +  cos  x  sin  ?/  cos  y 
dx  3  x^?/^  cos"-^  y  —  sin  x  —  cos^  y 

14.  Differentiate:    (1)  ^^1^^  +  log  VT^^^' ;   (2)  tan-^  ^^'^  ~  ^"'^^'"^; 

^Jl  —x^  •  a  +  h  cos  X 

(3)   cos-i^  +  ^^Q«^;        (4)    sin-i^JlA^ilL^;        (5)    tan-i  ^^' "  ^' ^^"-^; 

a  +  6  cos  X  «  +  6  sin  X  6  +  a  cos  x 

(6)    v'wsin--^x  +  ncos-^x;        (7)    (2a^  +  x*)'^a^  +  x*  ;        (8)   l^iBJ^^; 

(cosmx)" 

(9)   (cosx)«in^;    (10)  tan-i  ^^  +  ^''  +  V 1  -  x'^^ 

Vl  +  X=^  -  Vl  -  QC^ 

[Answers  to  Ex.  14:    (1)      ^^""^^    ;     (2)  _v^MZ«i  ;     (3)     ^«^  "  ^% 
L  ri-x2^^  6  +  acosx  a  +  6cosx 

(^      V^EH;         (5)     ^^^'  ,         (&-,\(m-n^ giL^^ 

a  +  &  sin  X  a  +  6  cos  x  2  Vm  sin2  x  +  n  cos2  x 

x«.     Wa  +  SVx    .      ,j.v  WW  (sin  nx)"»-i  cos  (mx  -  nx)  ,      .q.   rr.n<s'»•^8 
^^^       /_V-T T'     ^^^  (cosmx)-'  '      ^^^   ^'^'"^^ 

(cos2  X  log  cos  X  —  sin2  x)  ;     (10) 


VT 


^•] 


QUESTIONS  AND  EXERCISES.  383 


CHAPTER   V. 

1.  If  the  equation  of  a  plane  curve  be  y  =  0(a;),  find  the  equations  of  the 
tangent  and  the  normal  at  any  point,  and  find  the  lengths  of  the  tangent, 
normal,  subtangent,  and  subnormal. 

2.  Deduce  the  equation  of  the  tangent  at  the  point  (x,  y)  on  the  curve 
y  =f(x),  when  the  curve  is  given  by  the  equations  x  =  (p(t),  y  =  ^//(t). 

X 

Prove  that  -  +  ^  =  1  touches  y  =  6e  «  at  the  point  where  the  latter  crosses 
the  y-axis.        ^ 

3.  Find  an  equation  for  the  normal  at  any  point  on  the  curve  whose 
equation  is  /(x,  y)  =  0. 

4.  At  what  angle  do  the  hyperbolas  x'^  —  y^  =  a^  and  xy  =  b  intersect  ? 
Draw  sets  of  these  curves,  assigning  various  values  to  a  and  b. 

5.  Find  the  angle  of  intersection  between  the  parabolas  y"^  =  4  ax  and 
x^  =  4:  ay. 

6.  Find  an  expression  for  the  angle  between  the  tangent  at  any  point  of 
a  curve  and  the  radius  vector  to  that  point.     Show  that  in  the  cardioid 

r  =  a(l  +  cos  6)  this  angle  is  —  H — 

7.  Determine  the  lengths  of  the  tangent,  normal,  subtangent,  and  sub- 
normal, respectively,  at  any  point  of  each  of  the  following  curves  :  (1)  the 

X  X 

hyperbola   b'^^  -  a^y'^  =  o^-b'^ ;    (2)   the  catenary  y  =  -  (e«  +  e"«)  ;    (3)    the 

parabola  y"  =  ^x.  \_Ansi.    (1)    —  V(a2  _  x'^){a>  -  e^x?-),  -  Va*  -  e'-x:\ 

ax  a^ 

^in^,  ^;  (2)  y'        ,   t,   -^y— ,   ^Vi^^^r^;    (3)  10,7J,8,4^] 

X  a?-  y/yi  _  ^2     a      ^yi  _  a-2    a 

8.  Show  that  all  the  points  of  the  curve  y^  =  4:a(x  -\-  a  sin  -  )   at  which 

V  «/ 

the  tangent  is  parallel  to  the  axis  of  x  lie  on  a  certain  parabola. 

9.  (1)  In  the  curve  r=a  sin^-,  show  that  <(>  =  ix//.     (2)  In  the  lemnis- 

o 


cate  r'^  =  a^  sin  2  d,  show  that  i^  =2  6,  <f)  =  Sd,  subtangent  =  a  tan^  0  Vsin  2  d. 

10.  Solve  the  following  equations  :  (i)  4  x^  -{-  48  x'^  +  I6b  x  +  115  =  0  ; 
(ii)  9  x*  +  6  x3  -  92  x2  +  104  X  -  32  =  0  ;  (iii)  16  x^  +  104  x*  +  73  x^  -  277  x^ 
-  161  X  +  245  =  0. 

11.  Show  that  the  condition  that  ax^  +  3  bx"^  +  3  ex  +  cZ  =  0  may  have 
two  roots  equal  is  (be  —  ad)^  =  4  (ac  —  b^){bd  —  c^). 


384  INFINITESIMAL   CALCULUS. 

12.  Prove,  geometrically  or  otherwise,    that   provided   f(x)    satisfies  a 
certain  condition  which  is  to  be  stated 

f{x  +  h)  -fix)  =hf>(x  +  eh), 
where  ^  is  a  proper  fraction.     Show  that  it  is  possible  that  in  this  relation  d 
may  have  more  values  than  one. 

13.  If  A  is  the  area  between  the  graph  of  f(x),  the  x-axis,  a  fixed  ordi- 

fj  A 
nate,  and  the  variable  ordinate /(cc),  show  that  ^^=/(x). 

dx 

CHAPTER   VI. 

1.  Find  the  ?ith  derivative  of  the  product  of  two  functions  of  x  in  terms 
of  the  derivatives  of  the  separate  functions. 

2.  Find  the   fourth  derivative  of  x^  cos^  x   and  the  ?ith  derivatives  of 

1  t'^ 

(1)  x^  cos  ax  ;    (ii)  x*  cos*  x  ;    (iii)  tan-^  - ;    (iv)  sin^  x  cos^  x  ;    (v) ; 

(vi)  e«^  sin  bx.  ^  x^  -  I 

3.  Show  that 

(i)  D>»f^U(-i)^?i(^±ll.i:i-(^  +  ^^-^)^;  (ii)i>»(x^-ilogx)=(^-^)'; 

(iii)    2>/i/Lzl^\  =  ?IziI1!L?LJ;  (iv)  2)Xesinx)  =  _gsm*cosxsina;(sinx  +  3). 

\1  +  Xj         (1  "T  X)'^ 

4.  If  X  =  a(l  -  cos  0,  2/  =^  a(nt  +  sin  t),  then  ^  =  -  ^L^-S>^±±1. 

dx^  a  sin^  t 

6.  Derive  the  following  :  (i)  If  gy +  »;«/- e=0,  D^  u  =  y  •  (^  -  y)e^  -h^x^ 
(ii)  If  a;*  +  ?/*  +  4  a^a;?/  =  0,  (y^  +  a2a;)3^|  =  2  a^xy{xh/  +  3  a*).  (iii)  If 

da;2  (Aa;  +  by  +  fY 

6.    Prove   the   following:      (i)    If    ?/ =  sin  (m  tan-' a;),     (l+a;2)2^  + 

2 x(l  +  a;2)  ^  +  wi2^  =  0.       (ii)  Ii  y  =  {x  +  Vx2~^=T)«,  (a;^  _  l)  ^^  +  a:-^  - 


da:  ,,  TO      dx?        dx 

d^y 

dx^ 


7i^y=0.    (iii)  If:</2  =  sec2a;,  ?/+^=3«/5.     (iv)  If  ?/  =  (!  + xO=^  sin  (wtan-i?:), 


(1  +  a'O^  -  2(?n  -  l)a;^^  +  m(m  -  V)y  =  0. 

7.  If  aey  +  fte-^  +  ce=«  —  e-^  =  0,  determine  a  relation  connecting  the  first, 
second,  and  third  derivatives  of  y. 


CHAPTER   VII. 

1.  Write  a  note  on  the  turning  values  of  functions  of  one  variable. 

2.  Assuming/(x)  and  its  derivatives  to  be  continuous  functions,  investigate 
the  conditions  that /(a)  should  be  a  maximum  or  a  minimum  value  of  /(x). 


QUESTIONS  AND  EXERCISES.  385 

3.  Show  how  you  would  proceed  to  find  the  maximum  and  minimum 
values  of  a  single  variable,  and  to  discriminate  between  them, 

4.  If  f{x)  have  a  maximum  or  minimum  value  when  x  =  a,  and  f(x)  be 
continuous  at  x  =  a,  prove  that  f'(_x)  must  vanish  when  x  =  a.  Show  by 
means  of  a  diagram  that  the  converse  is  not  necessarily  true.  Examine  the 
case  in  which /(x)  has  a  maximum  or  minimum  value  when  x  =  a,  and/'(x) 
is  discontinuous  when  x  =  a. 

5.  If  x^  -1-  3  x-y  +  4  2/3  =  1,  show  that  y/\  is  the  maximum  and  that  I  is 
the  minimum  value  of  y,  where  x  can  have  all  possible  values. 

6.  ABCD  is  a  rectangular  ploughed  field.  A  person  wishes  to  go  from 
J.  to  C  in  the  shortest  possible  time.  He  may  walk  across  the  field,  or  take 
the  path  along  ABC ;  but  his  rate  of  walking  on  the  path  is  double  his  rate  of 
walking  on  the  field.     Show  that  he  should  make  through  the  field  for  a  point 

on  ^O  distant  h ^  from  C,  a  and  h  being  the  length  of  AB  and  BC 

respectively.  ^^ 

7.  Prove  that  the  greatest  distance  of  the  tangent  to  the  cardioid 
r  =  a(l  +  cos  ^)  from  the  middle  point  of  its  axis  is  a\/2. 

8.  AB  is  a  fixed  diameter  of  a  circle  of  radius  a  and  PQ  is  a  chord  per- 
pendicular to  AB ;  find  the  maximum  value  of  the  difference  between  the  two 
triangles  APQ^  BPQ  for  different  positions  of  the  chord  PQ. 

9.  Show  that  the  point  on  the  curve  4  ay  =  x^,  which  is  nearest  the  point 
(a,  2  a),  is  the  point  (2  «,  a). 

10.  Show  that  the  minimum  value  at  which  a  normal  chord  of  the  ellipse 

—  +  ^  =  1  recuts  the  curve  is  tan-^ . 

11.  Prove  that  the  greatest  value  of  the  area  of  the  triangle  subtended  at 
the  centre  of  a  circle  by  a  chord,  is  half  the  square  on  the  radius  of  the  circle. 

12.  A  slip  noose  in  a  rope  is  thrown  around  a  square  post  and  the  rope  is 
drawn  tight  by  a  person  standing  directly  before  the  vertical  middle  line  of 
one  side  of  the  post.     Show  that  the  rope  leaves  the  post  at  the  angle  30°. 

13.  Show  that  the  maximum  and  minimum  values  of  integral  algebraic 
functions  occur  alternately. 

14.  (i)  Show  that  the  points  of  inflexion  on  a  cubical  parabola  y'^  = 
(x  —  a)2(x  —  h)  lie  on  a  line  3  x  +  a  =  4  6.  (ii)  Show  that  the  curve 
y(x^  +  a^)  =  a2(a  —  x)  has  three  points  of  inflexion  on  a  straight  line, 
(iii)  Show  that  the  curve  x^  —  axy  +  6^  _  q  has  a  minimum  ordinate  at 

X  =  -^-z ,  and  a  point  of  inflexion  at  (—  6,  0). 

^'2 


386  INFINITESIMAL   CALCULUS. 

15.  Find  where  the  following  curves  have  maximum  or  minimum  ordi- 
nates  and  points  of  inflexion  respectively  :  (i)  y  =  a;*  —  4  x^  —  2  a:'^  +  12  x  +  4  ; 

(ii)  y  =  xe=';    (iii)  y  =  xe-^ ;    (iv)  y  =  xe-^l  Ans.    (i)  x--  1,    1,    3, 

1  ±f  >/3;  (ii)  x  =  -2;  (iii)  x  =  1,  x  =  2;  (iv)  x  =  ±  -^,  x  =  0,  x  =  ±  V|.  I 

■\/2  J 

16.  Find  the  inflexional  tangent  of  the  curve  y  =  x  —  x^  +  x^.  \_Ans.  27  y 
=  18x  +  l.] 

17.  Show  that  :  (i)  The  cone  of  maximum  volume  for  a  given  slant  side 
has  its  semi- vertical  angle  =  tan-i  ^72;  (ii)  The  cone  of  maximum  volume 
for  a  given  total  surface  has  its  semi-vertical  angle  =  sin~i  \. 

18.  Show  the  march  of  each  of  the  following  functions  :  (i)  sin^x  cosx; 
(ii)  sin2x  — x;  (iii)  x(a  +  x)2(a  —  x)^. 

19.  Examine  the  following  functions  for  maxima  and  minima  : 

^.^    x{x^-\~)  ^..^  x^+lx+_ll        ^^  1-x  +  x^  l  +  x  +  x\ 

^  ^    X*  -  X2  +  1  '  ^     ^    X2  +  4  X  -f  10  ^      ^    1  +  X  -  X2  ^      M  -  X  +  X2  ' 


(v)  X  Vax  -  x^ ;  (vi)   (x  -  1)*  (x  +  2)3 ;  (vii)   (1  -f  x)2  -  (x  -  x^)  ; 

(viii)  secx  — x;      (ix)  sinx(l  +  cosx)  ;      (x)  asinx+6cosx;      (xi)  x-" ; 

(xii)  — —  Ans.    C'l)  Two  max.,  each  =  I :  two  min.,  each  =—  ^  ■ 

^     Mog  X  L  ^  ^  "  " 

(ii)  max.  =2,  min.  =§ ;    (iii)  min.  =f ;    (iv)   max.  =3,  min.  =^;   (v)  min. 

^  3j^^2  .     (vi)  min.  =  0,  max.  =  12*  •  9^  -  7^ ;     (vii)  max.  =  0,  min.  =  8 ; 
16  /- 

■»/r  1  

(viii)    sin  x= ;     (ix)  max.  =  1.299;     (x)  max.  =  Va^^  ;?2^  min.  = 

—  Va^  +  b'^ ',  (xi)  min.  for  x  =  - ;    (xii)  min.  =  e.  | 

CHAPTERS   VIII.,   IX. 

1.  What  is  meant  by  partial  differentiation  ? 

2.  State  precisely  the  restrictions  as  to  the  function  /(x,  y)  so  that  the 

theorem     "-^    =    ^ -'     may  hold,  and  prove  the  theorem. 
dxdy     dy  dx         22 
Show  that  if /(x,  y)  =  xy^  ~  ^  ,  the  theorem  does  not  hold  for x=0,  y=0, 
and  explain  why.  ^  -\-  y 

3.  Explain  the  meaning  of  a  partial  derivative.  In  what  sense  may  we 
logically  speak  of  the  partial  derivative  of  c  with  respect  to  a,  when  c  is  a 
function  of  a  and  6,  and  a  and  b  are  both  functions  of  x  ? 

4.  Prove  Euler's  theorem  for  a  homogeneous  function  4>  of  x,  y,  z  : 


QUESTIONS  AND  EXERCISES,  387 

5.  If  w  be  a  homogeneous  function  of  the  nth  degree  in  any  number  of 

variables  x,  y,  z,  •••,  then  x^  +  ?/^  +  z—  +  •••  =  nu. 
dx        dy        ds 

6.  Verify  that  A(d^l]  =  A(^\  in  the  case  of  each  of  the  following 
functions  :  sin  {x^j),  cos  (  f  ^^  V  'og  /^!±l!\    ^/^V 

7.  Verify  the  following :    (i)  If  w  ==  sin-i  -  +  tan-i  ^,     x  ^  +  ?/  ^  =  0. 

y  X       dx       dy 

(ii)  If  wzzz(4a6-c2)4  4^  =  -^^.    (iii)  If  z=xHsin-'^  ^  -  y^  tan-i^',    ^'^ 
_  ^     '  ac2      dadb     ^    ^  X     ^  y   Sx  dy 

-^.  (iv)  If  y  =J\y  +  ax)+<p{y-  ax),    in  general  ^  =  a"^  ^. 


x2+?/2                                                                                                              /               o  ^^2                  Qy-2 

(v)  If  M  =  log  ^^  +  2  tan-i  ^,  du  =  -^^  (y  dx-x  dy).  (vi)  If  w=tan-i  ^, 

x^-y                 y            x^-y^  x 

5^w      ,      1      3?f  5w  5^« 


+  T-^T^^^+2(^  +  ^  +  ^)^  =  0-  (v"i)  If  M=Vx3  +  2/3, 


5x  6«/  5^     1  —  u-  dx  dy  dz 


(??/  d^y 


8.   Verify  the  following:    (i)  If    fs  a^  +  2W^Y  =  fa^-|- l" 

^^         V      <^x        y  U^^V       \   dx       jdxdx^' 

Vd^'V       \dy        Jdy^      ^  ^       ^   ^-"^[dx^        ^j^\dxj         ^   ^^Uxdx^ 

and  2/  =  024.2^,  (^  +  1)^  =  ^^  +  ^2  +  20.  (iii)  If  ^  +  -1^^ 

^  ^  ^  dx^     dx  dx^  ^    ^       dx^     l-^x'^dx 

+ '^ =  0  and  X  =  tan  0,    ^  +  y  =  0.  (iv)  If  (a  +  6x)2  ^ 

(I+X2)2  '       (?02^^  ^     ^^  V      -r         ^     ^^2 

+  .4(a  +  6x)^  +  jBy  =  i^(x)    and   a  +  6x  =  e',    b"^^ -\- b  (A  -  b)  ^  +  By 
dx  dt^  dt 

=  fI^Lh^]  .     (v)  If  ^  _  sec  ^  cosec  6^  +  yn^  tan2  (?  =  0  and  x  =  log  sec  (?, 

CHAPTERS   X.-XIV. 

1.  Explain  and  illustrate  the  meaning  of  integration. 

2.  If  /(x)  be  finite  and  continuous  for  all  values  of  x  between  a  and  5, 
prove   that    lim„^7i  {/(«)  +  /(a  +  /i)  +  /(a  +  2  A)  +  ...  +  /(a  +  n-1  h)]  is 

0(&)  -  0(a),  where  h  =  ^-=-^  and  —  0(x)  =  /(x). 
n  dx 

3.  Explain  fully  how  it  is  that  the  area  included  between  a  curve,  the 
axis  of  X,  and  two  ordinates  corresponding  to  the  values  Xo  and  xi  of  x  is 

represented  by  the  definite  integral    \  ^ydx. 


388  INFINITESIMAL   CALCULUS. 

4.  Give  an  outline  of  the  reasoning  by  wliich  it  is  shown  that  the  area 
bounded  by  the  two  curves  y  =  (p(x)  and  y  =  \p{x')^  and  the  two  ordinates 

a;  =  a  and  x  =  6,  is  \   {<p(x')  — \l/(x)}dx. 

5.  Prove  Simpson's  or  Poncelet's  rule  for  measuring  a  rectangular  field, 
one  of  whose  sides  is  replaced  by  a  curved  line. 

The  graph  of  y  =  x'^  is  traced  on  a  diagram.  If  0  be  the  point  (0,  0)  on 
it,  Pthe  point  (10,  100),  and  Pilf  the  ordinate  from  P,  find  the  area  of  031 P 
cut  off  between  OM^  MP,  and  the  curve,  by  taking  all  the  ordinates  corre- 
sponding to  integral  values  of  the  abscissas,  and  applying  the  rule  you  adopt. 
Tell  exactly  by  how  much  your  calculation  is  wrong, 

6.  Show  how  to  find  the  volume  of  the  sui-face  generated  by  the  revolu- 
tion of  a  given  curve  about  an  axis  in  its  plane. 

7.  Find  the  area  cut  off  between  the  parabola  y  =  x^  and  the  circle 
x^-hy^  =  2. 

8.  Trace  the  curve  whose  equation  is  a^y"^  =  x^(a^  -  x^),  and  find  the 
whole  area  enclosed  by  it. 

9.  Show  that  the  area  included  between  the  curve  y'^(2  a  —  x)  =  x?  and 
its  asymptote  is  3  -Ka^. 

10.  Determine  the  amount  of  area  cut  off  from  the  circle  whose  equation 
is  x2  +  ?/2  =  5  by  a  branch  of  the  hyperbola  whose  equation  is  xy  —  2. 

11.  Trace  the  curve  ay  +  2  x{x  —  a)  =  0.  Find  the  area  of  the  closed  por- 
tion contained  between  the  curve  and  the  axis  of  x.  If  this  portion  revolves 
round  the  axis  of  cc,  find  the  volume  generated. 

12.  A  curved  quadrilateral  figure  is  formed  by  the  three  parabolas 
y2  _  9  ax  +  81  a?-  =  0,  y''-  -  4  ax  -I- 16  a^  =  0,  ?/2  -  ax  -|-  a^  =  o,  the  other  boun- 
dary being  the  axis  of  x.     Find  the  area  of  the  quadrilateral. 

13.  Show  that  the  volume  of  the  solid  generated  by  revolving  about  the 
X-axis,  an  arc  of  a  parabola  extending  from  the  vertex  to  any  point  on  the 
curve,  is  one-half  the  volume  of  the  circumscribing  cylinder. 

14.  Determine  the  curve  for  any  point  of  which  the  subtangent  is  twice 
the  abscissa  and  which  passes  through  the  point  (8,  4). 

16.  Write  the  equation  including  all  curves  that  have  a  constant  sub- 
normal. Determine  the  curve  which  has  a  constant  subnormal  and  which 
passes  through  the  points  (0,  K)^  (b,  k),  and  find  what  is  the  length  of  its 

constant  subnormal.     ^Ans.  hy"^  =  (k'^  -  h^)x  +  bh"^ ;  ^!^l^.'j 

16.  In  what  curve  is  the  slope  at  any  point  inversely  proportional  to  the 
square  of  the  length  of  the  abscissa  ?  Determine  the  curve  which  has  this 
property  and  passes  through  (2,  5),  (3,  1). 


QUESTIONS  AND   EXERCISES.  389 

17.  State  and  derive  the  rule  known  as  "integration  by  part.^/'     Apply 
it  to  find   j  X"  log  X  dx. 

18.  Show  that  if  the  integral  of  /(x)  is  known,  the  integral  of  f~^{x)^  the 
function  inverse  to  /(x),  can  be  found. 

19.  Show  how  to  integrate  /=  J^^\  where  f(x)  and  0(a;)  are  rational 

(f>{x) 

integral  functions  of  x,  and  give  some  of  the  standard  types  for  the  integrals 
on  which  the  value  of  /  may  be  made  to  depend.  Show  how  to  integrate  the 
fraction  when  the  equation  0(x)  =  0  has  repeated  imaginary  roots. 

20.  Show  that  if /(«,  v)  is  a  rational  function  of  u  and  i?,  /  x,  -%/ — '^^—  \dx 

ax  +  b  ^       ^cx  +  dj 

can  be  rationalised  by  means  of  the  substitution  — -i—  =  «". 

ex  -\-  d 

21.  What  is  meant  by  a  formula  of  reduction  for  an  integral  ? 
Investigate    formulas    of    reduction    for    the   following :    (i)     i  sin"'  6  dd 

in    which    m    is    an    integer;     (ii)    |  sin"' tf  cos"  ^  fW ;    (iii)    i    ~  =dx ; 

(iv)  I  X"  sin  X  dx. 

22.  Explain  how  it  is  that    J    cos2»+i  0  rf^  =  0. 
dx 

p)  Vrtx2'+  2  6x  + 


23.    Evaluate    ( ,  by   means   of    the  substitution 


y{x  —  p)  =  \/ax?-  4-  2  &x  +  c. 

24.    Evaluate  the  following  integrals,  and  verify  the  results  by  differentia- 

tion:      f^"'°"''^,       psin-iJIi^dx,       T— ^V'       f'^^^^'^^'  ' 
•^  (1  +  ^.2)1       ^0  ^a^x  J^  sin  e  cos3  ^       J,   ^^^^  ^ 

r d^ C  x^  dx  C         dx  C  dx 

J  a2  cos2  ^  +  62  sin  2  ^'       J  x^^  _  l'        Jx(3  +  4x5)3'        J  a  sin  x  +  sin  2  x' 

f  x*(«  +  x)*f?x,  f     ^^  +  ^ — dx,         Txs  tan-i  x  dx,  fe^^  sin2  x  dx, 

-'  J  x2  —  4  X  +  3  J  ^ 

/• dx /*    (x  4-  l)dx 

J  X  V-x2  +  5x-6'      J  Vx2  +  X  +  1 


x^  -    ^  +  ^ 


CHAPTERS   XV.,    XVI. 

1.  Find  an  expression  for  the  area  bounded  by  a  curve  given  in  polar 
coordinates  and  two  straight  lines  drawn  from  the  pole. 

2.  Show  how  to  find  the  length  of  the  arc  of  a  plane  curve  whose  equa- 
tion is  given  (i)  in  rectangular  Cartesian  coordinates,  (ii)  in  oblique  Carte- 
sian coordinates,   (iii)  in  polar  coordinates. 


390  INFINITESIMAL   CALCULUS. 

3.  Investigate  a  formula  for  finding  the  superficial  area  of  a  surface  of 
revolution  about  the  axis  of  x. 

4.  Trace  the  curve  f^  =  a'^  cos  3  6,  and  find  the  area  of  one  of  its  loops. 

5.  Show  that  in  the  logarithmic  spiral,  r  =  a^,  the  length  of  any  arc  is 
proportional  to  the  difference  between  the  vectors  of  its  extremities. 


6.  Find  the  area  of  the  curve  r  Va^  -\-  b^  =  (a^  +  62)  cos  6  +  a"^. 

7.  Find  the  surface  of  a  spherical  cup  of  height  h,  the  radius  of  the 
sphere  being  E. 

8.  Find  the  average  value  of  sin  x  sin  (a  —  x)  between  the  values  0  and 
a  of  the  variable  x. 

9.  Find  the  volume  bounded  by  the  surface  a/-  +  'v/-  +  'v/-  =  1  and  the 
coordinate  planes.  ^ 

10.  The  axis  of  a  cone  is  the  diameter  of  a  sphere  through  its  vertex ; 
find,  in  terms  of  its  vertical  angle,  the  volume  included  between  the  sphere 
and  the  cone,  and  examine  for  vvhat  angle  it  is  greatest. 

11.  Determine  the  areas  of  each  of  the  following  figures :  (i)  The  segment 
cut  off  from  the  parabola  y'^  =  4  ax  by  the  line  2x  —  Sy  +  4ia  =  0.     (ii)  The 

2  2 

curve  /-V  +  ^^y  =  1.      (iii)  The  evolute  of  the  ellipse  («x)^  +  (by)^  = 

(a2  _  b^)i.  (iv)  The  figure  bounded  by  the  ellipse  16  x"^  +  25  y^  =  400,  the 
lines  x  =  2,  x  =  4,  and  2y  -^  x  =  S.  (v)  The  curVe  (x^  +  y^y  =  a^x^  +  &V« 
(vi)  The  oval  y  =  x^  -\-  V(x  —  1)(2  —  x).  (vii)  The  loops  of  the  curve 
a^y^  =  x2(a2  _  x^).  (viii)  The  segment  of  the  circle  x'^  +  y^  —  25  cut  oif  by 
the  line  x-{-y  =  7.     (ix)  The  area  common  to  the  ellipses  b^x^  +  a-y^  =  a^b^, 

a^x'^  +  6V  ^  ^2^2.  Vjins.   (i)  I  a".        (ii)  |  -Kob.       (iii)  f  tt  ^^^  ~  ^'^^^ 

L  ab 

(V)     ^(«^  +  &2).  ^^i>)     ir_  ^^j.^     -gg^^j^    2(j2.  (viii)      2^5  sin-1  ^3  -  |. 


(ix)    4  a6  tan-ill 


12.  Find  the  volume  and  the  area  of  the  surface  generated  by  the  revolu- 
tion of  the  cardioid  r  =  a(l  —  cos  6)  about  the  initial  line.     [Area  =  -^^  7r«-.] 

13.  Show  that  the  volume  enclosed  by  two  right  circular  cylinders  of 
equal  radius  a  whose  axes  intersect  at  right  angles  is  ^^  a^,  and  the  surface 
of  one  intercepted  by  the  other  is  8  a^. 

14.  Show  that  the  volume  included  between  the  surfaces  generated  by 
the  revolution  of  a  hyperbola  and  its  asymptotes  about  the -transverse  axis 
and  two  planes  cutting  this  axis  at  right  angles  is  the  same,  no  matter  where 
the  sections. are  made,  provided  that  the  distance  between  the  planes  is  kept 
constant. 


QUESTIONS  AND  EXERCISES.  391 

15.  The  parabola  y'^  =6x  intersects  the  circle  x"^  -\-  y"^  =  16.  Show  that 
if  the  larger  area  intercepted  between  the  curves  revolves  about  the  ic-axis, 
the  volume  generated  is  60  w  cubic  units  ;  and  show  that  if  the  smaller  area 
intercepted  revolves  about  the  ?/-axis  the  volume  generated  is  ^i^VS^^  cubic 
units. 

16.  An  arc  of  a  circle  of  radius  a  revolves  about  its  chord.  Show  that  if 
the  length  of  the  chord  is  2  aa,  volume  of  the  solid  =  2  7ra3(sin  «  —  |  sin^  « 
—  a  cos  a),  surface  of  the  solid  =  4  7ra"^(sin  a  —  a  cos  a). 

17.  Find  the  area  of  the  segment  cut  off  from  the  semi-cubical  parabola 
27  ay^  =  4  (x  —  2  ay  by  the  line  x  =  6  a.  Also  find  the  volume  and  the  area 
of  the  surface  generated  by  the  revolution  of  this  segment  about  the  a^-axis. 

^Ans.     2^4  «2,  ^^2  I  Z^  _,.  3  log  (  V2  +  1)  I .] 

18.  A  number  n  is  divided  at  random  into  two  parts.  Show  that  the 
mean  value  of  the  sum  of  their  squares  is  f  n^. 

19.  Show  that  the  mean  of  the  squares  on  the  diameters  of  an  ellipse,  that 
are  drawn  at  points  on  the  curve  whose  eccentric  angles  differ  successively 
by  equal  amounts,  is  equal  to  one-half  the  sum  of  the  squares  on  the  major 
and  minor  axes. 

20.  Prove  that  the  mean  distance  of  the  points  of  a  spherical  surface  of 

radius  a  from  a  point  P  at  a  distance  c  from  the  centre  is  c  H or  a  -f  — , 

according  as  P  is  external  or  internal. 

CHAPTER   XVII. 

1.  Define  curvature  of  a  curve.  Find  an  expression  for  the  radius  of 
curvature  of  a  curve  whose  equation  is  in  the  form  y  =f(x). 

2.  Show  that  the  curvature  at  any  point  of  the  curve  given  by  x  =  ^(O, 

y  —  \p{t)  is  ^^ — ~r^    ^  where  accents  denote  differentiations  with  respect 
to  t.  (</)'-^  +  ^p'2)^ 

r 

3.  For  any  curve  /(r,  &)  =0  show  that  radius  of  curvature  = TTT' 

in  which  ^..tan-i^.  '^^'^V+d^J 

dr 

4.  Find  the  coordinates  of  the  point  on  the  parabola  x^  =Aay  for  which 
the  radius  of  curvature  is  equal  to  the  latus  rectum. 

5.  Show  that  at  a  point  of  undulation  the  tangent  has  contact  of  at  least 
the  third  order. 

6.  Show  that  the  circle  (4  a;  -3  a)2  -i-  (4  ?/  -  3  a)2i=  8  a^  and  the  parabola 
Vx+V^=Va  have  contact  of  the  third  order  at  the  point  [-,  -  j.  Find 
the  order  of  contact  of  the  curves  y  =  x^  and  y  =  3x'^  —  Sx+1. 


392  INFINITESIMAL   CALCULUS, 

7.  Show  that  the  circles  of  curvature  of  the  parabola  ?/2  =  4  ax  for  the  ends 
of  the  latus  rectum  have  for  their  equations  x^  +  ij^  —  10  ax±^ay  —  ^  cfi  =  0, 
and  that  they  cut  the  curve  again  in  the  points  (9  a,  T  ^  a). 

8.  Find  the    radius    of    curvature   of   each    of    the    following    curves : 

(i)  The  cardioid   r^  =  a^  cos  ^  d.     (ii)  ij  =  2x  -\-  Sx'^  -  2  xtj  +  y'^  at  (0,  0). 

(iii)  xy'^  =  a\a  +  :«)  at  (  -  a,  0).     (iv)  The  tractrix  x  =  a  log  cot  -  —  a  cos  6, 

z 
y  =  asm  6.     (v)  ?/  —  x  —  sin  x  at  the  origin,  and  where  x  =  - •    (vi)  The  expo- 
X  2 

nential  curve  y  =  ae^.     (vii)  r""  =  a'^cosmd.      (viii)  r=:asinw^  at  (0,0). 
(ix)r^=a^cos3e.     \  Ans.     (i)  fVar.     (ii)  |\/5.     (iii)  |  a.      (iv)-«cot^. 

(v)0,2V2.     (vi)  i^'  +  y')\     (vii) ^ -.     (viii)^na.     (ix)  ^.] 

^^  ^    ^  cy  ^     ^(w+l)/""-!      ^      ^^  ^    ^41-2  J 


CHAPTER   XVIII. 

1.  Define  an  asymptote  to  a  curve.  Derive  a  method  of  finding  the 
asymptotes  of  an  algebraic  curve  whose  equation  in  Cartesian  coordinates  is 
of  the  nth  degree. 

2.  Show  that  the  asymptotes  of  the  cubic  x^y  —  xy^  +  y'^  +xy  -{-  x  —  y  =  0 
cut  the  curve  again  in  three  points  which  lie  on  the  line  x  -\-  y  =  0. 

3.  Find  the  asymptotes  of  the  curve  xy^  -  x^  -{-  2  x^  +  3  »/  +  .x  —  1  =  0. 
Show  that  the  points  at  a  finite  distance  from  the  origin  in  which  the 
asymptotes  cut  the  curve  lie  on  the  line  3y  +  2x  —  1=0. 

4.  Draw  the  curve  x/^y  =  x^  —  a^.  Show  that  it  has  an  asymptote  which 
crosses  the  x-axis  at  an  angle  tan~i  3. 

5.  Find  the  asymptotes  of  the  following  curves :  (i)xy^—x^y  =  a^(x-\-y)-\-h^. 

(ii)  1  4-  2/  =  c^.  (iii)  x^  -  xy^  +  ay-  -  a^y  =  0.  (iv)  (x'^  +  ?/2)  (y^  _  4  y.2^ 
+  4  y^(x  -  1)  +  x2(4  X  +  3)  =  0.  (v)  (x  -  2  a)tf' =  x^  —  a\  (vi)  x^  +  3  y'^ 
=a^(y-x).  (vii)  x3+2x2|/+xi/2-x2-x?/  +  2  =0.  (viii)  ?' sin  2  ^  =  a  cos  3  ^. 
(ix)  y^  =  x'^(2a-x). 

6.  Find  the  asymptotes  of  the  curve  x^y  —  xy^  +  6  a^xy  4-  o,^y  — 16  a^x  =  0. 
Show  that  the  origin  is  a  point  of  inflexion. 

7.  Define  a  family  of  (plane)  ciirves,  and  the  variable  parameter  of  the 
family.  Define  the  envelope  of  a  family  of  curves.  Define  an  ultimate 
intersection  of  a  family  of  curves.  Define  the  locus  of  the  ultimate  intersec- 
tions of  a  family  of  curves.  Illustrate  the  definitions  by  concrete  examples 
and  diagrams,  and  furnish  any  explanations  you  may  think  necessary. 

8.  Show  that  in  general  the  locus  of  ultimate  intersections  of  the  family 
touches  each  member  of  the  family.  Show  that  this  locus  is,  in  general,  the 
envelope  of  the  family.  Explain  the  necessity  of  the  qualifying  phrase  "  in 
general." 


QUESTIONS  AND  EXERCISES.  393 

9.    Explain  the  method  of  finding  the  envelopes  of  the  curves /(a;,  ?/,  t)=0, 
where  « is  a  variable  parameter. 

10.  Write  a  note  on  "singular  points  of  curves,"  explaining  what  they 
are,  giving  illustrations,  and  showing  how  to  find  them. 

11.  Ellipses  of  equal  area  are  described  with  their  axes  along  fixed  straight 
lines.     Show  that  the  envelope  consists  of  two  equilateral  hyperbolas. 

12.  Prove  that  the  circles  which  pass  through  the  origin  and  have  their 
centres  on  the  equilateral   hyperbola  x^  —  y^  =  cfl  envelop  the  lemniscate 

13.  P  is  a  point  on  a  parabola  of  which  A  is  the  vertex.  Find  the  equa- 
tion of  the  curve  touched  by  all  circles  described  on  ^P  as  diameter. 

14.  A  circle  passes  through  the  origin,  and  its  centre  lies  on  the  parabola 
y2  —  4  dx.     Show  that  the  envelope  of  all  such  circles  is  a  cissoid. 

15.  A  straight  line  moves  so  that  the  product  of  the  perpendiculars  on  it 
from  two  fixed  points  (±  r,  0)  is  constant  i~  ^•2).     Show  that  its  envelope  is 

the  ellipse  — f-  —  =  1,  or  the  hyperbola ^  =  1. 

16.  Find  the  envelope  of  circles  passing  through  the  centre  of  an  ellipse 
^2^2  _|.  ^2^2  _  ^252  and  having  centres  on  the  circumference  of  the  ellipse, 
[Ans.  (a:2  +  1,2)2  ^  4(^2^2  4.  52^2).] 

17.  Ellipses  are  described  having  their  axes  coincident  in  direction  with 
those  of  a  given  ellipse,  and  lengths  of  axes  proportional  to  the  coordinates  of 
a  variable  point  on  the  given  ellipse.  Show  that  the  ellipses  all  touch  four 
straight  lines. 

18.  Find  the  equation  of  the  envelope  of  the  line  a:sin «  +  ?/cos«  = 
a  sin  «  cos  a. 

19.  From  a  fixed  point  on  the  circumference  of  a  circle  chords  are  drawn, 
and  on  these  as  diameters  circles  are  drawn.  Show  that  the  envelope  of  the 
series  of  circles  is  a  cardioid. 

20.  If  a  cannon  is  fired  at  an  elevation  6,  and  the  projectile  has  an  initial 
velocity  equal  to  that  attained  by  a  body  in  falling  h  feet,  the  equation  of  the 
parabolic  path,  referred  to  horizontal  and  vertical  axes  through  the  point  of 

projection,  is  y  =  xtRn6 sec2^.     Find  the  envelope  of  the  paths  for 

different  elevations. 

CHAPTERS   XIX.,  XX. 

1.   A  function /(x)  is  defined  by  an  infinite  series  f(x)  =  ^  0n(ic)  ;  state 

»i=i 

and  prove  a  sufficient  condition  that  the  equation  — f(x)  =  ^  —  <t>n{^)  nniay 
be  true.  ^  S^^ 


394  INFINITESIMAL    CALCULUS, 

2.  Write'  a  note  on  the  conditions  under  which  (1)  the  integral,  (2)  the 
differential  coefficient  of  an  infinite  series,  may  be  obtained  by  integrating  or 
differentiating  the  series  term  by  term. 

3.  Prove  that  if /(x)  be  a  continuous  function  of  x,  then 

f{x  +  h)  =  f{x)  +  hf'{x  +  eh), 
where  0<^<  1. 

Show  clearly  how  this  proposition  may  be  applied  to  prove  Taylor's  theo- 
rem, and  specify  the  circumstances  in  which  the  theorem  as  you  state  it  is  true. 

4.  Prove  Taylor's  theorem  for  the  expansion  of  f(x  +  h)  in  ascending 
powers  of  h,  carefully  specifying  the  conditions  which  f(x)  must  satisfy. 
Find  an  expression  for  the  remainder  after  n  terms  of  the  series  have  been 
written  down. 

5.  State  Maclaurin's  theorem,  and  give  the  conditions  under  which  it  is 
applicable  to  the  expansion  of  functions.     Derive  the  theorem, 

6.  Expand  in  series  of  ascending  powers  of  x  the  functions :  (i)  cos  mx. 
(ii)  tan-i(«4-x).  (iii^sin(?7isin-i  x).  (iv)  (1  +  ?/)*,  where  y<,l. 
(v)  e"«^  +  e-"*\     (vi)  e^^^+A,  4  terms. 

7.  Expand  the  following  functions  in  powers  of  x  :  (i)  e^i^  ^.     (ii)  tan-i  x. 

(iii)  cot-la;.  IAiis.    (i)   1  +  x  +  |  x^  -  i  x*  -  ^V  x^  +  •••.  (ii)   For 

values  of  x  from  x  —  —  1  to  x  =  I,  x  —  ^  x^  -\-  I  x^  —  ^  x"^  +  -"  ;   for  |  x  |  >  1, 

---  +  ^: —  +  -'.      (iii)  For    |xl<l,    ^- x  +  i  x^  -  ix^  +  ...  ;    for 

2x8x35x5^  ^    ^  ''^'2  '  ^ 


X     3x3     5x5  J 


X     3  x^     5  X 
8.    Calculate  the  values  of  the  following 


xdx. 


(i)    i   x^Vl  —  x^dx.     (ii)   (   xcotxc?x.     (iii)    i      e""^  dx.     (iv)    \   e^sin 

(V)  ^l  ^  dx.  ]^Ans.   (i)  f  x^(l  -\x^-  ^h  ^'  -  ^V  ^'  +  -)• 

(ii)  x-^^-^-^....      (iii)  2(l+l+-l-  +  _l_  + I +...V 

^   ^         9     225     6615  ^    ^     V       3     1 -2 -5     1 .2 .8.7     1 -2 .3.4.9         J 

^^2!       3!        4!        6!         7!         8!  ^^         3.3!      5-51      '"j 


CHAPTER   XXI. 

1.    Solve  the  following  equations : 
(1)  x'^y  dx  -  (x3  +  y^)dy  =  0.  (2)  3  e^  tan  ydx-\-(l  -  e')  sec^  y  dy  =  0. 

(3)  (x^-ixy -2y'^)dx-^(y'^-4xy-2x^)dy  =  0.  (4)  xDy-y=xVx^-\-y'^. 
(5)  {x^  +  y^)(xdx  +  ydy)=a^(xdy-ydx).  (6)  (x^  +  l)Dy  ^  2xy  =  ix^. 
(7)  6(x  +  l)Dy  -y  -yK        (8)  p^  -  4  xyp  +  8  ?/2  =  0,    in   which  p  =  D^y. 


QUESTIONS  AND  EXERCISES.  895 

(IS)  D.^y  +  2D.^y  +  D.y  =  0.  (14)   g  _  3  g  +  4  |  -  2  2/ =  0. 

(20)  2/  §  +  i  (i  -  2  2/)  =  0.      (21)2  xD^ij  D^y  -i- a^  =  {D^yy, 

[Solutions :  (1)  3  y^  log  y=x^  +  c.     (2)  tan  ?/  =  c(l  -  e^y.     (3)  a:^  -  6  a;2«, 
-  6  a;y2  +  2/^  =  c.      (4)  2  y  =  x(ce*  -  ce-=*) .      (5)  a;2  +  ?/2  ==  2  a^  tan-i  ^  +  c. 


(6)  3(x2  +  i)y  =  4  a;3  +  c.       (7)  V^+ 1(1  -  y^)  =  cy^.      (8)  ?/  =  c(x  -  c)^. 

(9)  2  2/-5  =  0x5  +  5  x^  (10)  y  =  a:2(i  +  cgx).  (H)  (^^2  +  ?/)2(x2  -  2  y) 
+  2  x(x2  -Sy)c  =  c2.  (12)  1  +  2  ci/  =  c2x2.  (13)  y  =  Ci  +  e~^(c2  +  C3X). 
(14)  y  =  e*(ci  +  C2  cos  x  +  C3  sin  x).  (15)  y  =  Ci  +  C2X  +  e'^  (cs  +  C4X). 

(16)  x?/  =  ci  logx  -  log  (x  -  1)  +  C2.         (17)  y  =  X  (ci  +  Co  log  x)  +  C3X-1. 

(18)  sm(ci-2V2«/)=Coe-2x.    (19)  x=->/cj/2-y+     ^     hycos-J(2cy-l)  +  Ci. 

c  2cVc  ^ 

(20)  2  X  =  log(!/2  +  ci)  +  C2.     (21)  15  Ci2?/  =  4(cix  +  a2)t  +  CoX  +  Cg.] 

2.    Find  the  singular  solutions  of : 
(1)  x2p2_3  xyp+2  y^-{-x^=0.     (2)  xp2-2  yp  +  ax=0.     (3)  Solve  equation  (2). 
r Solutions :   (1)  x^iy^-ix^)  =  0.     (2)  y^  =  ax\     (3)  2  1/ =  cx2  +  ^-l 

MISCELLANEOUS. 

1.  How  far  does  the  symbol  —  obey  the  fundamental  laws  of  algebra  ? 

dx 

2.  Prove  that  if  D  denote  — ,  and  /(Z>)  be  any  rational  algebraic  func- 
tion of  2),  i\\enf{D)uv  =  uf{D)^  +  Diif'(D)v  +  ^  f"(D)v  +  —. 

3.  If  <p  denote  any  function  of  x,  prove  that  ^!(^  z=  n^I'^ -\-  x^- 
By  this  theorem  or  otherwise  find  the  value  of  D'^^x  sin  mx). 

4.  K   x  =  e»,    prove  that  -^f-^- iV^- 2 V"f—-w  +  lV  =  a^"^i 

de\dd       J\dd       J     \dd  I  dx« 

f  d       d  \^        /fZ\""/d\** 
where  u  is  any  function  of  x.    Prove  also  that  f  —  x  —  |  if=   —  )  x    —  I  w. 

\(7x     cZx/  \f7x/       \(?x/ 

5.  If  0(x)  is  a  function  involving  positive  integral  powers  of  x,  prove  the 
symbolic  equation  </>  f—  (  e«^  •  ?n1  =  e''''<t>(a  -\ jw. 

6.  Show  how  to  find  the  values  of  -^  and  —^  when  x  and  y  are  con- 

dx  dxr 

nected  by  the  equation  /(x,  y)  =  0. 


896  INFINITESIMAL   CALCULUS. 

7.  If  M  =/(aj,  y)  and  if  x  =  (p(t),  y  =  \p{t),  state  and  prove  the  rule  for 
obtaining  the  total  derivative  of  u  with  respect  to  t. 

If  x  =  r  cos  ^,  y  =  r  sin  d,  transform  (x'-^  —  w^)  - — —  -\-  xy    -r-^  —  ^—5     into 

an  expression  in  w^hich  r  and  6  are  the  independent  variables. 

8.  Calculate  the  nth  derivative  of  (sin~ia;)2.     Show  by  the  use  of  Mac- 
laurin's  theorem  that  (sin-i  x)2  =  2  —  +  -  —  +  ^—^jL  + 

9.  The  curves  u  =  0,  u'  =  0  intersect  at  (x,  y)  at  an  angle  a.    Show  that 

d^t  du[  _  du^  du 
doc  dy      dx  dy 


tan  a 


du  du__  ^  drii^  d_u 
dx  dy      dx  dy 


y2  ift.  ffl  r.fl 

Show  that  the  curves  —Ar    ,  =  1  and  ^—  +  ^  1=  1  intersect  at  right  angles 

10.  Show  that  the  total  surface  of  a  cylinder  inscribed  in  a  right  circular 
cone  cannot  have  a  maximum  value  if  the  semi-angle  of  the  cone  exceeds 
tan-i  \,  i.e.  26°  34'. 

11.  Through  a  diameter  of  the  base  of  a  right  circular  cone  are  drawn  two 

planes  cutting  the  cone  in  parabolas.     Show  that  the  volume  included  between 

4 
these  planes  and  the  vertex  is  —  of  the  volume  of  the  cone. 

3  TT 

12.  Calculate  the  area  common  to  the  cardioid  r  —  a{\  —  cos  ^)  and  the 
circle  of  radius  f  a  whose  centre  is  at  the  pole. 

13.  Find  the  area  and  the  perimeter  of  the  smaller  quadrilateral  bounded 
by  the  circles  tP-  ■\-  y'^  ■=.  25,  x^  -f  ^2  _  ^44^  ^.^(j  ^j^g  parabolas,  y^  _  g  x, 
yi  +  12  (X  +  2)  =  0. 

14.  Given  the  cardioid  r  =  4  (1  —  cos  0)  and  the  circle  of  radius  6  whose 
centre  is  at  the  cusp,  find  the  length  of  the  circular  arc  inside  the  cardioid 
and  the  lengths  of  the  arcs  of  the  cardioid  which  are  respectively  outside  the 
circle  and  inside  the  circle. 

15.  If  a  curve  be  defined  by  the  equations  -— "=:  — ^  = ,  find  an  ex- 

0(0     V'CO     /(O 
pression  for  the  radius  of  curvature  at  a  point  whose  parameter  is  t. 

16.  Expand  (by  any  method)  x^  cosec^  x  in  a  series  of  powers  of  x  as  far 
as  the  term  in  x*.  At  what  place  of  decimals  may  error  come  in  by  stopping 
at  this  term,  when  x  is  less  than  a  right  angle  ? 

17.  Trace  the  curve  x*  -1-  ?/*  =  d?-xy,  and  find  the  points  at  which  the  tan- 
gent is  parallel  to  an  axis  of  coordinates.     Find  the  area  of  the  loop. 

18.  Trace  the  curve  x  =  a  sin  2  ^ (1  +  cos  2  0),  1/  =  a  cos  2  ^  (1  —  cos  2  &). 
(a)  Prove  that  Q  is  the  angle  which  the  tangent  makes  with  the  axis  of  x,  and 
obtain  the  equation  of  the  tangent  to  the  curve.  (6)  Find  the  length  of  the 
radius  of  curvature  in  terms  of  Q. 


QtJt^STlONS  AND  EXERCISES.  897 


19.    Find  ^  under  each  of  the  following  conditions  :    {\)  x^  =  e 
dx 


(ii)  y  =  e*""  tan-i  x.    (iii)  e''  -{■  x  =  €«  +  y.    (iv)  y  = •    (v)  sin  (xy) 

-e^y-x'^y  =  0.  x  +  Vl-x^ 

20.  Four  circles  x^-{-y^  =  2  ax,  x^  +  y'^  =  2ay,  x'^-\-y^  =  2  bx,  x~  +  y^  =  2  hy, 
form  by  their  intersections  in  the  first  quadrant  a  quadrilateral ;  prove  that 

the  area  of  this  is  (a-  +  6')  cot-i    ^^  ^^  .^  -  (a  -  6)2. 

21.  Prove  that  the  area  of  a  sector  of  an  ellipse  of  semi-axes  a  and  h  be- 
tween the  major  axis  and  a  radius  vector  from  the  focus  is  —  (0  —  e  sin  0), 

where  0  is  the  eccentric  angle  of  the  point  to  which  the  radius  vector  is 
drawn. 

22.  Trace  the  curve  xy^  =  a^  ;  and  find  whether  the  area  between  it,  a 
given  ordinate,  and  the  coordinate  ajces  is  finite. 

Show  also  that  if  the  tangent  at  P  meet  the  axis  of  x  in  T,  then  MT=SOM, 
where  3/ is  the  foot  of  the  ordinate  at  P,  and  O  is  the  origin. 

23.  If  w  be  a  homogeneous  function  of  n  dimensions  in  x  and  y,  show  that : 

(i):«-^f^+2xj,/l'-+y-^f^=»(,.-l)«.       (ii)xfi^+,/|-  =  («-l)f. 
dx-  dxdy       dy-  dx^       dxdy  dx 

(iii)  x^  +  ypi^  =  (n  -  1)  f .        (iv)   (x^  +  yj-W  =  »%. 
dxdy       dy^  dy  V  dx       dy) 

24.  Prove  the  following  :  (i)  If  ?<  =  sin-^  {xyz),    d^djidu^  ^an2  u  sec  u. 

dxdy  dz 

(ii)  If  u  =  log  (tan  x  +  tan  y  +  tan  z),  sin  2  x^  -\-  sm2  y^  -\-  sm2  z^  =  2. 

dx  dy  dz 

(iii)  If  u  =  log (x^  +  y^  +  z^-S  xyz),   ^  -H  ^  -h  ^  =  —-1-—.      (iv)  If 

dx      dy       dz      x-\-y  +  z 

u  ^  tan2  X  tan2  y  tan2  z,  du  =  4  m  f    '^^     +     ^■'     +     ^^    ^  • 

\  sin  2  a:      sin2y     sin2  zj 

25.  If  b  be  the  radius  of  the  middle  section  of  a  cask,  a  the  radius  of  either 
end,  and  h  its  length,  show  that  the  volume  of  the  cask  is  -^-^  tt  (3  a2  ^  4  ab 
-t-  8  b'')h,  assuming  that  the  generating  curve  is  an  arc  of  a  parabola. 

26.  031  is  the  abscissa,  MP  the  ordinate  of  a  point  P(xu  y\)  on  the 
hyperbola —  =1,  (a^i,  2/i,  hoth  being  positive).    If  A  is  the  vertex  nearest 

P,  show  that  area  AMP  =  \  Xiyi  —  ^  «&log  ( ^  -h  ^  ),  and  area  sector  OAP 

27.  Show  that  the  mean  of  the  squares  on  the  diameters  of  an  ellipse  that 
are  drawn  at  equal  angular  intervals  is  equal  to  the  rectangle  contained  by 
the  major  and  minor  axes. 


398  INFINITESIMAL   CALCULUS. 

28.  Find  the  mean  square  of  the  distance  of  a  point  within  a  square  from 
the  centre  of  the  square. 

29.  Through  a  diameter  of  one  end  of  a  right  circular  cylinder  of  altitude 
h  and  radius  a  two  planes  are  passed  touching  the  other  end  on  opposite  sides. 
Show  that  the  volume  included  between  the  planes  is  (tt  —  ^)d^h. 

30.  Show  that  the  integration  of  the  expression  /(x,  y)dxdy  may  be  per- 
formed in  any  order,  provided  the  limits  of  x  and  y  are  indepen'dent  of  each 
other. 

31.  Evaluate  111  x"-y^zy  dx  dy  dz  taken  throughout  the  space  bounded 
by  the  coordinate  planes  and  the  plane  x  -\-  y  -\-  z  =\. 

32.  Prove  geometrically  or  otherwise  that  x  dy  —  ydx=r'dd,  and  show  that 
the  area  of  a  closed  curve  is  represented  by  ^  j  (x  dy  —  y  dx) . 

33.  The  equation  to  a  curve  being  written  in  terms  of  the  polar  coordi- 
nates r  and  d,  p  being  the  perpendicular  from  the  pole  to  the  tangent  and 

u=-,  show  that,  -=u^-^  f  —  ^  '^ 
r  p^  \dd , 

34.  If  a  is  a  first  approximation  to  a  root  of  the  equation  f(x)  =  0,  deter- 
mine graphically  or  otherwise  the  conditions  under  which  a  —  ^y^  is  a  valid 
second  approximation.  ^  -^  ^^^ 

35.  If  /(x)  be  a  finite  and  continuous  function  of  x  between  x  =  «  and 
X  =  b,  show  that  a  value  Xi  of  x,  lying  between  a  and  b,  may  be  found  such 
that/'(xi)  =  {f(b)  -f(a)}  -^{b-  a). 

If  the  function  be  x^'-l-cx,  find  the  point  in  question  when  a  =  a  and  &  =  2  a, 

and  thence  show  that  in  this  case  Xi  is  such  that  ^  ~    ^  is  constant  for  all 
values  of  a.  o  —x\ 


36.   Find  the  radius  of  curvature  of  the  curves:  (i)  lima^on  r='a  cos  d-\-b^ 
r  =  ^\  (ii)a2/2=(x 


where  r  =  -\  (ii)  ay"^  =  (x— a)(x  —  by  at  (a,  0).     Trace  the  curves.       Ans. 


,  (ii)  ^^-^n 


37.    (1)  Trace  the  curve  r=a-t-&  cos  0,  a>6>0  ;  find  its  area.     (2)  Find 
the  area  of  the  loop  of  y^  =  (x  —  1)  (x  —  3)2.     (3)  Find  the  area  between  the 

X-axis  and  one  arch  of  the  harmonic  curve  y=b  sin  -•       Ans.   \{2  a^-\-V^)Tr., 

,-  -,  a      i- 

32  V2 


15 


',  2a6.] 


38.  Trace  the  curve  ^y"^  —  (x  -\- 7)(x  +  4)2.  Find  the  area  and  the  length 
of  the  loop,  ^nd  the  volume  and  area  of  the  surface  generated  by  the  revolu- 
tion of  the  loop  about  the  x-axis.     [A7is.   |  V3,  4V3,  f  tt,  Sir.'] 


QUESTIONS  AND  EXERCISES.  399 

39.  Find  the  limiting  values  of :  (i)  log  7^/^"/  ,  when  (?=7r ;  (ii)  f  l^i^\i, 

(ir'^  —  ff^)d  \    X    J 

when  a;  =  00  ;  (ill)    — ^"  "~  ^ ,  when  x  =  I  ;  (iv) ,  when 

1  —  X  4-  log  X  2  x^     2  ic  tan  r  x 

1^ 

x  =  0;  (V)   /?i^V^  when  X  =  0  J  (vi)   ^^:=-^ ,  when  x  =  0  j  (vii)    ^—Z^"^ 
\   X    J  X  x^  —  a- 

when  x  =  a. 

40.  Find  the  mass  of  an  elliptic  plate  of  serai-axes  a  and  &,  the  density 
varying  directly  as  the  distance  from  the  centre  and  also  as  the  distances  from 
the  principal  axes. 

41.  From  a  fixed  point  A  on  the  circumference  of  a  circle  of  radius  a,  the 

perpendicular  ^F  is  let  fall  on  the  tangent  at  P.     Prove  that  the  greatest 

3"\/3 
area  APY  can  have  is  a^. 

42.  A  rectangular  sheet  of  metal  has  four  equal  square  portions  removed 
at  the  corners,  and  the  sides  are  then  turned  up  so  as  to  form  an  open  rec- 
tangular box.  Show  that  the  box  has  a  maximum  volume  when  its  depth  is 
\(^a  -\-b  —  y/d^  —  ab  +  b'^),  a  and  b  being  the  sides  of  the  original  rectangle. 

43.  Two  ships  are  sailing  uniformly  with  velocities  u,  v,  along  straight  lines 
inclined  at  an  angle  6 :  show  that  if  a,  b,  be  their  distances  at  one  time  from  the 
point  of  intersection  of  the  courses,  the  least  distance  of  the  ships  is  equal  to 

(av  —  bu)  sin  6 

(u^  +  v''^  -  2  uv  cos  e)^ 

44.  A  right  circular  conical  vessel  12  inches  deep  and  6  inches  in  diameter 
at  the  top  is  filled  with  water  :  calculate  the  diameter  of  a  spherical  ball  which, 
on  being  put  into  the  vessel,  will  expel  the  most  water. 

45.  A  statue  a  feet  high  is  on  a  pedestal  whose  top  is  b  feet  above  the  level 
of  the  observer's  eyes.  How  far  from  the  pedestal  should  the  observer  stand 
in  order  to  get  the  best  view  of  the  statue  ?     [Ans.   y/b{a  -\-  b)  feet.] 

46.  The  lower  corner  of  a  leaf,  whose  width  is  a,  is  folded  over  so  as  just 
to  reach  the  inner  edge  of  the  page :  find  the  width  of  the  part  folded  over 
when  (1)  the  length  of  the  crease  is  a  minimum,  (2)  when  the  area  of  the  tri- 
angle folded  over  is  a  minimum.     [^Ans.  (1)  fa;  (2)  fa,] 

47.  (1)  Show  that  the  cylinder  of  greatest  volume  for  a  given  surface  has 
its  height  equal  to  the  diameter  of  the  base,  and  its  volume  equal  to  .8165  of 
that  of  the  sphere  of  equal  surface. 

(2)  Show  that  the  cylinder  of  least  surface  for  a  given  volume  has  its 
height  equal  to  its  diameter,  and  its  surface  equal  to  1.1447  of  that  of  the 
sphere  of  equal  volume. 


400  INFINITESIMAL   CALCULUS. 

48.  Trace  the  graph  of  y  ^si"  2  x  -  sin  a;^    -p^^^  ^^^  2iY\g\es,  at  which  it 

cos  X 
crosses  the  a;-axis,  and  show  that  its  finite  maximum  distance  from  the  a;-axis 

is  (2i  -  l)t. 

49.  An  ellipse,  whose  centre  is  at  the  origin  and  whose  principal  axes  coin- 
cide with  the  axes  of  x  and  y,  touches  the  straight  line  qx+py=pq  ;  find  the 
semi-axes  when  the  area  of  the  ellipse  is  a  maximum,  and  also  the  coordinates 
of  its  point  of  contact  with  the  given  line. 

50.  Find  the  volume  of  the  greatest  parcel  of  square  cross-section  which 
can  be  sent  by  parcel  post,  the  Post-ofiBce  regulations  being  that  the  length 
plus  girth  must  not  exceed  6  feet,  while  the  length  must  not  exceed  3  feet 
6  inches. 


INTEGRALS. 

FOR   EXERCISE   AND   REVIEW. 

The  following  list  of  integrals  provides  useful  exercises  in 
formal  differentiation  and  integration.  It  will  also  afford  some 
assistance  in  the  solution  of  practical  problems  as  a  table  of  refer- 
ence. Those  who  have  to  make  considerable  use  of  the  calculus 
will  find  it  a  great  advantage  to  have  at  hand  Peirce's  Short  Table 
of  Integrals*  (Ginn  &  Co.). 

GENERAL  FORMULAS   OF   INTEGRATION. 

Formulas  A,  B,  C,  pages  173,  174  ;  formula  for  integration  by  parts, 
page  177. 

FUNDAMENTAL  ELEMENTARY   INTEGRALS. 

Formulas  L-XXVL,  pages  172,  173,  180,  181.  (These  should  be  mem- 
orised.) 

REDUCTION  FORMULAS  FOR    (x^^^^a  +  bx^ypdx. 

[Here  X denotes  (a  +  ftx").] 

1.  (x'^XP dx  =  ^"•""^^^^'  -  "CM-n  +  D  C^m-nxP ax. 
J  h{np -\- m  ■\- \)     b(np  +  in-\-  1)J 

2.  (x^XP  dx  =  a^"^^^Xi>+^  _  Hm  +  n  +  np  +  l)  C^m^nxP  dx. 

J  aCm  +  1)  .  a(i»-l-l)         •^ 

3.  ix^XPdx=    «^*"^^^^    +        ^^P        (x^XP-^dx. 
J  m-{-  np  +1     in-\-  np  +  1  •/ 

4.  ix'-XP  dx  =  -  oc«'*'XP*^  ^m  +  n  +  np  +  1  C^„.j^ph-i  ^^. 

J  an(p-\-l)  an(p-{-l)     J 

*  There  are  two  editions,  the  briefer  edition  of  32  pages  and  the  revised 
edition  of  134  pages. 

401 


402  INFINITESIMAL   CALCULUS. 

J  b7i(p  +  l)       bn(p  +  l)J 

J  771  -\- 1       m  +  IJ 

7  r   ^a; 1 (m  -  n  ■{■  np  —  l)b  r     dx 

J  x"'Xp~      {m  -  l)ax"'-iXp-i  (w  -  l)a        J  x"»-«X^ 

8  T-^^^  —  1 ,  m  —  n  +  np  —  \  C     dx     _ 

J  x'^Xp  ~  an(p  -  1)x'»-iXp-i  an{p  -  1)      J  tC^JTp-i' 

9  (XPdx  _  _         Xp+1         _  6(m  -  ?i  -  wp  -  1)  fX^c^a; 

10     fr^^  = Z^ I  «J_^? CXP-^dx^ 

J     x^         {np  -  m  +  l)a:'»-i      «j9  —  wi  +  1  J       x"* 

-J      rx"*c?x  _  x"'-"+^ a(m  -  »i  +  1)  rx'^-^t^x^ 

J    Xp       6(w  -  «p  +  \)Xp-^      h(jn  -  wp  +  1)  J      Xp 

12     r?^L^ x"*+i w  4-  n  —  np  +  1  rx"*(?x 

'    J    Xp  ~  an{p  -  \)Xp-^  an{p  -  1)      J  Xp-^' 

13.     f ^ = ^- r -  +  C2n-3)f ^ ]. 

J  (a  +  6x2)"     2(71  -  l)a  L(a  +  &x2)«-i     ^  J  {a  +  hx^y-U 

Put  a2  for  a,  6  =  1,  and  compare  with  Ex.  3,  Art.  118. 

14  r      ^^^^      — Zl^ 1  C         dx 

J  (a  +  &x2)«~2  6(n-l)(a  +  6x2)«-i     2  6(w  -  1)  J  (a  +  6x2)"-i' 

15  r___^___  =  1  r  dx hC       dx 

J  x\a  4-  6x2)«     a  J  x\a  +'  hx^y-^     aJ  {a-\-  hx^y  * 


EXPRESSIONS  CONTAINING  Va  +  hx. 
Also  see  Ex.  10,  page  191. 
C        dx        _     Va  +  bx      b    C       dx 

J  nri-^/n  _L   h^r  ttX  2aJr 


16. 


x2Va  +  6x  «^  2a^xVa  +  6x 

17.     f^^«+5idx  =  2V^Tfei  +  af— ^^ 

J        X  J  xy/a  +  bx 


INTEGRALS.  403 

EXPRESSIONS  CONTAINING  Vx2  ±  dK 
Also  see  Ex.  7,  page  191. 

18.  f^g^  =  logp  +  ^^±«"'V    See  XXIV.,  XXV.,  page  181. 

n 

19.  r(a:2  ±  a2)2-^^  =  ^(^±ji!}i  ±  Ji«L  r(x2  i  a4~'dx. 
J  n  +  1  71+lJ 

20.  f  (a;2  ±  a2)^dx  =  -  VW^cfi  ±  —  log  (x  +  Vx^To^). 
•/  2  2 

21.  f  (x2  ±  a2)|^a;  =  |(2  x2  ±  5  a^) y/x^  ±  a^  +  3a_*  iog(x  +  >/x2  ±  a?-). 
Jo  8 

22.  f  x2(x2  ±  a2)^  dx  =  I  (2  x2  ±  a2)  Vx2  ±  a2  -  ^  log  (x  +  \/x2  ±  a2) . 
•/  8  8 

23.  f ^ dx  =  ± ^ 


(x2  ±  a2)f  a2Vx2  ±  a2 

2^     C      x^dx      ^  X  ^^2  _j_  ^2  :p  ^  log  (^a;  ^  Vx2  ^  a2). 
•^  (x2  ±  a2)i     2  2 

25.  f'    a;2(?x      ^_         x        j^i^^^^  j^^^i  _  ^^x 

•^(x2±a2)t         Vx2-a2 

26.  r ^ =  llog ? ;     f ^ =  lsec-i?. 

*^x(x2  +  a2)i     «       a  +  Vx2  +  a2     ^a:(x2-a2)i     «  « 


27.     r         ^3;  ^  -^  Vx2  d,  a2 

•^  x2(x2  ±  a2)^  «^a; 


I     a    C         ^x  ^      Vx2  +  a2        1         g  +  Va;2  +  a2 

■^x3(x2  +  a2)^  2a2x2        2a3 

6.  r ^^__  =  ^^^E«:%  J_sec-i?. 

•^x3(x2_a2)^        2a2x2        2a^ 

>.    a.  I  ^ ~ =  Vx2  ^  a^  -  a  log  — ■ ■ 

J  X  ^  X 

(X2  -  a2)i  clX 


V  x2  —  a^  —  a  cos-i  -  • 

X 


6.   f(^^ 

«/  X 

J  X'^  X 


404 


INFINITESIMAL   CALCULUS. 


EXPRESSIONS   CONTAINING  y/a^  -  x^. 
Also  see  Ex.  7,  page  191. 


31.     f  (g2  -  x^ydx  =  ^^^'  ~  f )'  +  -^  f(a2-a;2)2  'djc. 

J  W  +  1  W+lJ 


X2 


34 


35 


J  Va2  -  x2  »^  m        J  Va2  _  x'-^ 

^  m  +  2  W  4-  2  J   y'^2  _  jp2 


<^»^  dx  =  —       "^^^  ~  ^^       '       »'*  -  2 


J 


\/a2_x2 


-  x2       ^       m  -  2      r  <?x 


dx  =  — 


1     m  -  2  J  3 


x"*  (m  —  2)x"*-^     m  —  2  ^  x*"  Va^  —  x^ 

2 


36.  f  (a2  _  a:')^ (?x  =  -  A/a2  -  x2  +  —  sin-i  ?. 
J  ^  2  2  a 

37.  f  (a2  _  x2)' dx  =  ^ (5  a2  _  2  x2)  Va2  -  x2  +  — -  sin-i  ■ 
JO  8  ( 

38.  j" x2(a2  -  x2)*  to  =  -  (2  x2  -  a^)  Va^  -  y?-  +  ^  sin-i  2 

39.  J. 


X2  fZx  X 


(a2  -  x2) 
40.    I        '^'^ 


Va2  _  x2  +  ^  sin-i  ^^ 
i         2  2  a 


(«2_;;c2)f      a2Va2_a;2 


J- 


x2dX 


(a2  _  x2)^      V«^  -  a;2 

i.     r ^^ ^_Va2_a,2^     ^g      /^ aj, ^Ij^g X 

•^x2(a2-x2)^  ^"^^  *^x(a2_a;2)i    /^       a  +  Va2-x2 

•J 


^'5^^^^+j_,„g 


a«(a2-x2)*  ^""^^        "^"^       a+Va2_a:2 


45     fiai^^^^V5f3F^-alog«+^^2i^^. 

^  X  X 

46.   ri«izL^^.-_:^^«IE?- 

J  X'' 


INTEGRALS.  405 


EXPRESSIONS   CONTAINING  V2  ax  -  x\   V2  ax  +  x^. 


[Here  X  denotes  V2  ax  —  x^,  and  Z  denotes  V2  ax  +  x^. ] 
47.    a.  f^  =  sin-i^^l«.  6.  r^=  log  (x  +  a  +  Z). 


48.    a.  rX(?x=^ — «X  +  — sin-i 
J  2  2 

6.  fZ(?x  =  ^^t«Z-^log(x+a  +  ^. 

J  m+2  m+2J 

6.  f^^Zd^  ^gli^^- (2 '"  +  1)''  f^-iZdx. 
J  m+2  m+2J 

50.    a.  f-^  = ^^ +      ^-^       f    ^^     . 

Jx'^X         (2?)i-l)ax'«      (2  w  -  l)a  J  cC^-iX 

'  J  x'^Z"  (2  m  -  l)ax'"      (2  m  —  l)a  J  x'»-iZ 

■    ^'  J     X  ?)i  m  J       X     ' 

,     rx"*dx_x'"-^Z      (2  m  -  l)a  rx"^-^(fa 
'  J     Z  m  m  J       Z 

62.    ..f^d.  = ^ +  ^n:^rx^_ 

J  x"*  (2m-3)ax'«      (2  m  -  3)a  J  x'"-i 

b.  C^dx  =  -—^ ^^-3       C   ^    dx. 

J  x"*  (2  m-  3)ax"*      (2  m  -  3)a  J  x*"-! 

53.  a.  f  xXdx  ^  _  3  ^^  +  ^x  -  2  x^  ^     og  ^.^_,  x-a, 

J  6  2a 

6.  fxZdx=:-^^'-^^-^^'z  +  ^log(x  +  a  +  Z). 

54.  a.  f^^  =  -^.       6.f^  =  _^. 

^  xX         ax  J  xZ         ax 

^    ^    r^  =  _X+asin-i5-=^.        6.  r^  =  Z  -  alog(x  +  a  +  Z). 
J   X  a  J    Z 


''  X 

56.    a.  l'^!i^  =  -^±l«X  +  ^a2sin-i^i:^. 


X  2  2 


6.  f^  ^  ^  -  3  Q'  z  +  I  aMog  (x  +  a  +  Z) . 


406  INFINITESIMAL   CALCULUS. 

57.  a.  r^^  =  X+asin-i^^li?.       h.  (^^  =  Z  +  alog(x+ a  +  Z). 

J     X  a  J     X 

58.  a.  C^dx  =  -^-sm-^^^-^:^'      b.  C^dx  =  -^+log(x-\- a  + Z). 

J  x^  X  a  J  x^  X 

69.    a.  i^dx^-^^'       h.  (^dx  =  ~-^. 
J  x3  3  ax3  J  x^  3  ax3 

61     /7    fx  f^x  _    X  ■,     Cxdx  _  X 

'  J'X^~  aX  '  J  Z^~  aZ 


EXPRESSIONS  CONTAINING  a  +  hx±cx'^. 

i.    a.  \ =  — z==  tan  i  -,  for  b^<i4:ac 

J  a  +  bx-^  cx2      V4  ac  -  62  V4  ac  -  b'^ 

—  log ■ ,  for  62  ;>  4  a,c. 


V62  -  4  ac        2  ex  +  6  +  V62  -  4  ac 

■  J  a  +  6x  -  cx2      V62  +  4  ac        V62  +  4  ac  -  2  ex  +  6 

63.  a.  r  <^^     —  ^  J_  log  (2  ex  +  6  +  2  Vc  Va  +  6x  +  cx2). 

*'  Va  +  6x  +  cx2      Vc 

6.  f  "-^  =  4 sin-    ^'^^-^   . 

•^  Va  +  6x  —  cx2      Vc  V62  +  4  ac 

64.  a.  f  Va  +  6x  +  cx^dx  =  ^  ^^  "^  ^  Va  +  6x  +  cx2 

J  4e 

-  ^'^  ~  ]  ^^  log  (2  ex  +  6  +  2  Vc  Va  +  6x  +  cx2). 
8c^ 

6.  rVa  +  6x-c.x2(^x  =  g^^:::^Va+6x-cx2  +  ^^^±i^sin-i    ^^^-^   • 
-^  4  c  g  J  V62+4  ac 


gg     ^    I  X(fx  _  Va  +  6x  4-  ex2 


i.   a.  J 


Va  +  6x  +  cx'^  ^ 

^  log  (2  ex  +  6  +  2  Vc  Va  +  6x  +  cx2). 

2c^^ 

^    r  xc?x  _  _  Va  +  6x  -  cx2      _6_  ^.^.j    2  ex  —  6 

^  Va  +  6x  -  ex2  c  ^  ^^  VPTToc 

N.B.    Other  algebraic  integrals  that  are  occasionally  useful  are  given 
in  Exs.  7-10,  page  191,  and  in  Exs.  4,  6,  page  222. 


66 


INTEGRALS.  407 


EXPONENTIAL  AND  TRIGONOMETRIC   EXPRESSIONS. 

The  most  elementary  of  these  are  given  in  the  integrals  on  pages  172, 180. 

a.  fsinxcos^xt^x^-^^^^^.  6.  fsin»a:cosa;  =  ^*^"^^^ 

J  n+  1  J 


n-\-l 

67.  a.  fsin2xdx  =-- ^sin2x  b.   f  cos2xdx  =|+ ^sin  2x. 

CO      r  ■         ■,  sin«-ixcosx  ,  n  —  I  C  •  „  9    j 

68.  \  sm'^xdx  = h \  &n\^-^xdx. 

J  n  n     J 

«rt     C  ,       cos^-i^sinx     n  —  lC  o    -, 

69.  I  cos^xdoj  = 1 I  cos"-2x(^a:. 

J  n  n     J 

»t.     C   dx  1        cos  X     ,  n  —  2  C    dx 

70.  \ = —  H I  -: — -z-  * 

J  sm'^x         n  —  1  sin"-ix      n  —  \J  sm"-2/j; 

w-i      C   d^    _      1         sin  X        n  —  2  C     dx 

J  cos«x     n  —  \  cos^-ix     n  —  1  J  cos^-^^ 

72.  Csecnxdx  =  ^^'^  ^  '^""'"'^  +  ^^-=^  fsecn-^xdx.     (Cf.  71.) 

J  W  —  1  ?l  —  IJ 

73.  fcosecnxdx  =  -^^^-^^^^^^^^  +  ^Lzi2rcosec»-2xc?x.     (Cf.  70.) 
J  n  —  1  n  —  IJ 

74.  f  tan«  x  dx  =  ^H1!L2^  _  ftan^-s  x  dx. 

75.  f  cot**  X  dx  =  -  ^2i!!li£  _  f  cot«-2  x  dx. 
J  ?i  —  1        ^ 

76.  f sm"» a; cos » x tta;  =  - ^i-""' '^^ «»s»+i a; 
V  ill  +  /i 

+  ^~^  f sin»»*- 2 a; cos** x dx. 
ni  +  nJ 

77.  rsin-«.cosn^^^..«m!!^il^J5^ 

J  IW  +  1 


78.  Jsln".  X  COS"  »;<?»!  =  ?'""'^*  '^  *»^"-  *  '^ 


,      fi  —  1 

79.  Jsln-  a;  cos"  xd^  =  ^J"™^*  '^  *«^"*'  ^ 


f  sin»w  a?  cosw-2  05  <?«?. 


n  +  1 


+  ^^  +  ^  +  2  f  sin»»*  a;  cos»»+2  a?  cTa?. 
w  + 1      ^ 


408  INFINITESIMAL   CALCULUS. 

80.  fsin  mx  sin  nxdx=-  ^^"  (^  +  ^)^  +  ^i"  (^  "  ^)^. 

81.  rcosmxcosn:Kdx=     sm  (m  +  n)x     sin  (m  -  n^ 
J  2  (m  +  «)  2  (w  -  w) 

82.  fsin  mo;  cos  nx  dx  =  -  ^^^  (^  +  ^)^  -  ^"^  (^^ "  ^)^. 
J  2  (m  +  w)  2  (m  -  n) 

83.  f ^ =        ^        tan-i  f  a/^^  tan  ^^ ,  when  a>b 

J  a-\-bcoBx     y/a^  —  b^  \^a  +  b        2) 

y/b  +  a  +  Vb-a  tan  - 
—  log ,  when  a<b. 


Vb^-a^        y/b  +  a-Vb-a  tan  | 
a  tan  -  +  6 


84.     f ^ =        ^    -^  tan-i  —     ^  when  a  >  6 

J  a  +  6  sm  x      Va2  _  52  Va^  _  52 


atan-+&-\/&2_a2 
log ,  when  a<6. 


v^^^^a^      atan5+6+V62_a2 
2 


85.  f ^ =  J_tan-if^i?^V 

J  Or'  cos2  ic  +  52  sin2  X     ab  \     a     ) 

86.  f  e«^  sin  nx  dx  =  e"''(«siQ  »^^  -  ^  cos^^;;^     ^g^^  ^^  ^g  ^^.^  ^^g  x 
^  a^  +  ^a  ^  '  '' 

87.  fe-cosnxdx=2^1^?^i5^^^i^^^^i^.     (See  Ex.  6,  Art.  106.) 


r 
\ 

\ 

57? 

\ 

\ 

2 

/ 

+1 

/ 

f 

1 

-1 

V 

\ 

\ 

IT 

> 

\ 

■z 

y 

( 

'  % 

X 

I 

\ 

-TT 

y-'Snix 


y  —  coax 


TtFX 


y^sinz^x        y -cos'^x 


409 


1 

J 

y 

y 

TT 
~2 

0/ 
/ 

1 

1 

TT 

y=ta 

n  a; 

/"       ¥ 

X 

27r 

37r                     __ 

^     ^ 

TT 

t-^ 

^^-^^^ 

Jt 

'^^^ 

t/=tan 

_87r 
I- 

'a?' 

410 


"X 


i/  =  sec'a; 


411 


The  Parabola  x^-v  y^  =a'' 


The  Cubical  Parabola  o?y=x'^ 


The  Astroid  or  Four-Cusped 
Hypocycloid,  a;  ^  + 1/  ^  =  a  ' 


Asymptote 


The  Cissoid  of  Diodes 
^3 


The  Witch  of  Agnesi 

l/  =  x2-|-4o* 


412 


The  Folium  of  Descartes 


O  X 

The  Catenary 


Asymptote     ^O  X 

The  Exponential  Curve 


The  Cycloid 
a;=a  (^-sln^),i/=a  (1-cos  ^) 


O  a       X 


0\  X 


The  Logarithmic  Curve 
l/=log  X 


Parabola 
r=asec-  ^ 


The  Cardioid 
r=a{l-cos6) 


413 


The  Lemniscate,  rLa^ cos 2 0,       The  Curve,  7- a  sin  2^     The  Parabolic  Spiral 


Asymptote 


The  Spiral  of  Archimedes,  r=a(j 


The  Hyperbolic  or  Reciprocal 
Spiral,  r  0  =^  Cb 


X 


The  Lituus  or  Trumpet,  Tlie  Log-arithmic  or  Equiangular 


r'  6  -^a' 


Spiral,  r-e**^  or  log  r-a  <? 


414 


ANSWERS   TO   THE   EXAMPLES. 


(6)  2x4-1;    (c)  3x2;    (rZ)  ^;    (e)  :=;^ ;    (/)  ^;    (^)  ^;    (A)  -^ 


>>«<< 


CHAPTER   I. 

Art.  4.  1.  45°,  0°,  63^  26' 4",  71°  33' 54",  75°  57' 49",  78°  41' 24" 
80°  32' 16",  82°  52' 30",  104°  2' 11",  99°  27' 44",  135°,  126°  52'.2,  110°33'.3 
2.   (.18,  .033),  (.29,  .83),  (.5,  .25),  (.87,  .75),   (5.72,  32.66),   (-1.07,  1.15) 

(-  .35,  .12),  (-  .18,  .033),  (-  .09,  .008).      3.    [The  latter  part.]     (a)  -- 

y  \by  \by  y  a^y 

fi)  9J^.       4.    a.  CO,   ±  .5774,   ±  .2582,  0,   ±  .4045,  ±  1.8074  ;  90°,  30°  and 

150°,  14°28'.7  and  165°31'.3,  0°,  22°  1'.4  and  157°58.'6,  61°2'.7  and  118°57'.3. 
b.  27,  12,  3,  0,  6.75,  18.75;  87°  52'.7,  85°  14'. 2,  71°33'.9,  0°,  81°34'.4. 
86°56'.8.  c,  00,  ±  1.4142,  ±  1,  ±  .8165,  ±  .5774,  ±  .5,-  90°,  54°44'.l  and 
125°  15'.9,  45°  and  135°,  39°  14'  and  140°  46',  30°  and  150°,  26°  34'  and  153°  26'. 

d.  0,  i.l937,  ±.4330,  oo,  ±.0945,  ±.3034;  0°,  10°57'.7  and  169°2'.3, 
23°24'.8  and   156°35'.2,   90°,    5°  24'    and   174°  36',    16°  52'.7  and   163°7'.3. 

e.  00,  ±.8661,  ±.8183,  ±1.25,  ±.9139;  90°,  40°  53'.8  and  139°6'.2, 
39°  17. '6  and  140°  42. '4,  51°20'.4  and  128°39'.6,  42°25'.4  and  137°34'.6. 
5.    Where  x  =  ±  2.57  ;  where  x  =  ±  2.78. 


CHAPTER   IT. 

Art.  12.      1.   35.2426   or   26.7574,    29.9586   or  28.0614,    SVsinx  + -^ 

1J.      r^  sinx 

+  7sin2x  +  2.     2.   68,  28,  14,  3  sin2x  -  5  sinx +  21.     3.    ^^  ~  '^  ^.     4.    18  + 

2-49X 

8\/x  +  X,  4  +  \/x2  +  2.     5.    ay^  +  bxy  +  cx2,  (a  +  6+  c)x%  (a  +  6  +  c)2/2. 

CHAPTER   III. 
Art.  22.     4.    (a)  2x,  2  x,  2x;    (6)  3x2,  33.2,  3x2.        5.   4^.3^  2  x  +  4, 

-  -,    -  4  -  3  +  4  X.       6.   6  ^  12  ^2  _  8  -  ^.       7.   6  2/6,   §  «  _  8  +  ^. 
x2         x2  '  ^2  ^  '   2  ^  y2 

Art.  26.  2.  2  wr  times,  r  being  the  measure  of  the  radius  ;  1.51  sq.  in. 
per  second  ;  2.83  sq.  in.  per  second.  3.  .866  a  times,  a  being  the  measure 
of  the  side  ;  25.98  and  51.96  sq.  in,  per  second.  4.  4  irr'^  times,  r  being  the 
measure  of  the  radius  ;  9.425  and  37.7  cu.  in.  per  second.     5.  5l\  mi.  per  hour. 

415 


416  INFINITESIMAL    CALCULUS. 

Art.  27,     3.   Sx'^dx,  dx,  2dx,  Sdx,  adx,  2xdx,  lixdx,  etc.         4.    1.6; 
1.681.     6.  42.2  ;  43.696.     Ex.  5.03  and  9.425  sq.  in.     Ex.  1.3  and  2.6  sq.  in. 

CHAPTER   IV. 

Art.  31.     6  x2  +  14  X  -  10,  2  X  -  17,  -  2  x  +  21. 
Art.  32.     4.   (5  X*  -  8  x3  +  21  x2  +  2  X  -  2)  dx,  .... 

Art   33      1    3  X*  -  14  x3  +  6  x^  16x-21x-^-  x^       -  2  x^  +  44  x  -  96 

(8x2-7x  +  2)2'  (x3  +  8)2       '       (2x-^-9x  +  3)-'i  ' 

(3  X*  -  14  x3  +  6  x'-^)  dx    ^          2    ^  _  J^     -_8 

(3  x-^  -  7  X  +  2)-2       '  ""        ■    '^'  640'    245' 

Art.  35.     2.  ii-Ciii:^.       3.  ^ 


4  M-  17  3  X  +  7 

Art.  37.    1.  2  w^\  12  «3^— ,  63  i<8— ,  8  x^  12  x^,  84  x",  27  x2  -  34  x  +  10. 

dx  dx  dx 

3.   240  x(5  x2  -  10)23,  120  x3(3  x*  +  2)9,  (432  x^  +  300  x^  -  168  x^  +  448  x  -  50) 

(4  x2  +  5)7(3  x*  -  2  X  +  7)4.      4.-2  21-^  u\  -  7  ir^  u\  -  11  m-^'^  w',  _  7  x-s, 

-15X-6,    -170X-11,    -8x(x2-3)-5,    -60x3(3x4+ 7) -6,    15  x*- 21x2  + 

-  -  ^  +  -•     5.    A  u^ Du,  -  f  zr* Du,  I J Du,  h x'K  ^- x^,  f  x'",         ^^       , 
x2      x3      x*  o  .  .        .         V3x2-5 

^^  +  '^(2x2  +  7x-3)"3,     1 ,     _9(3x-7)~^,    6x-|x"^-x~^- 

3  V2X  +  7 

2  x~i  +  /s  x"i  6.    V2  2/2-1  ^^'^   V3  x^3-i^  5 V7  x^7-i,  2 V5(2  x  +  5)^^-1^ 

\/3(6  X  +  7)  (3  x2  +  7  X  —  4)^3-1,       7.  —  +  c,  and  give  c  any  three  particular 

4 

constant  values.     8.   (In  each  of  these  expressions  k  is  to  be  given  any  three 

/y6  1  2^  2^  62 

particular  constant  values. )     — \-  k, h  k,   ~-x^  -\-  k,   -x'^  +  k,   -  x^  +  - 

6  X  3  5  5  X 

-2Vx  +  A:.      12.    6x2  +  34x-61,  max"»-i- n&x-'*-!         ^^  ~"^^ 


(1  -  x2)2'  (a  +  x)2 
_12  +  5^-f_35^4, i « ,       _1^^, 

^'        3  '  X2V1  +  X2  (a_&x2)2  (l_x2)^ 

,      mnx«-i(l  +  x«)"»-i,      12  6x2(a  +  6x3)3,      x^-\l  -  x)"-i 

•     (I  -X)\/1-X2 

[m-(m  +  .)x],     ^-'^^.  14.    a.  «|^^^  4f^f±^^' 

2V^^::^  2/2 -ax    a(3  2/2-2x2) 

9x2y-  8x-  14xy2-2y3     -(x  +  a)y2 ^   _  x^   _6^. 

14x2?/  +  6x?/2_3x3- 16?/   (a  +  ?/)(&2_aj/_2«/2)  +  ?/(x  +  a)2'       y        a2y 

&0  -  f ,  I,  I,   -  f .       17.    ?/  =  x2  +  ^',  in  which  k  is  an  arbitrary  constant ; 

?/  =  x2  +  1.       18.    5  ft.  per  second.       19.    10  ml.  an  hour  ;  8f  ft.  per  second. 

20.   (4,8).         21.    3hr.  ;60mi.         22.    f  ft.  per  second.         23.    36°52'.2. 

24.    36°52'.2. 


ANSWERS,  417 

Art    39      1     (6a;  +  4)logae  6x  +  4  .4.34(6x  +  4)  11 

•       •    3x2  +  4a;-7'     Sx^  +  ^x-f     3x^-\-^x-f     16  log,  a' 

ii,  .29868.       2.    i  .144765.       3.   -=^^,    -1-,     L_^,    -— i=, 

1  _  a;2     1  _  a:^      (1  -  a:)  Vx      v^M^ 

— —,  1  +  log  X.       4.    log  (x2  +  3  X  +  5)  +  c,  log  c  (x3  -  7  X  -  1),  log  Vkx, 
xlogx 

in  which  c  and  ^  are  arbitrary  constants.     (Ex.   Write  each  of  these  anti- 
derivatives  with  the  arbitrary  constant  involved  in  other  ways.) 
6     (a)    -  (21^7  +  1877  x  +  228  x^)  Vx  +  2  .^.  6(x^  -  2) 

30(4x-7)^(3x  +  5)t          '  (x+l)Hx  +  2)-^' 

.. ^91  x2  +  475  X  +  450 


Art.  41. 


15  (2  X  +  5)^  (7  X  -  5)^  (x  +  3)^ 

Art.  40.     1.    2xe^\    2.303  (10^,    2.303(6  x  •  10*'='),    -^e"^^-      2.    2  e^ 

2\/x 
2.303  (2  t .  10*'),  2  ^e''+3,  4.606  (102«+7),     3.  e'  x"-i  (x  +  wi).  ««'"  •  a;"-i  log  a, 

"t~-V'  ('-^>*-^'  (^^'  ^"(^-^)-    *•  i'"-^''   ^^"  +  ''' 
I  gSx+i  4-  c,  c  being  an  arbitrary  constant. 

L.    2.  (3x+7)««r2xlog(3x+7)+^|^1,  (3x+7)2xriog(3x+7)2 

+  3!^],  as  last,  Vi(l^^^|x-''.x-l(7^1ogx+l),6e'.e^  -^(^Vlogx, 

-  -  log  a. 

X 

Art.  42.     1.    —  sin  2  w  =  cos  2  ?<  •  —  (2  «)  =  2  cos  2  «  •  — ,  3  cos  3  u  -  Du, 
dx  dx  dx 

^  cos  I  M  .  m',  I  cos  f  w  — ,  Y  COS  J^  w  .  Du.       2.    D  sin  2  X  =  COS  2  X  •  i>  (2  x) 

=  2  cos  2  X,      3  COS  3  X,      |  cos  ^  x,      6  x  cos  3  x^,      3  sin  6  x,      20  x*  cos  4  x^, 
20sin4  4xcos4x.     3.  5cos5^  «cosi«2.     4    2  cos2  xsin  3  x-3  sin  2xcos3x 

^  sin2  3x 

sin2x  +  2xcos2x,    2  xsin  (x  + -W  x2cos  [x  + -V  5.    45°  and  135°. 

6.    Where  x  =  rnr  ±  •  9553,  in  which  n  is  any  integer.        7.   63°  26'  and  116° 

34'.        8.   Where  x  =  nir  -  -,  in  which  n  is  any  integer  ;  54°  44'.  1  and  125° 
4 

15'.  9  ;  where  x  =  wtt  +  j,  n  being  any  integer.        9.    n  cos  nx,  wx"-i  cos  x**, 

n  sin«-i  X  cos  x,      2  x  cos  (1  +  x^) ,       w  cos  (nx  +  a) ,       w6x"-i  cos  (a  +  6x") , 
12  sin2  4  X  cos  4  x,  ^^osx-sinx^   cos(logx)    ^^^      ^,  ^^^  j  smef . 

X2  X  X 

10.    (a)  sin  x  +  c,    i  sin  3  x  +  c,     ^  sin  (2  x  +  5)  +  c,     ^  sin  (x^  —  1)  +  c,    in 
which  c  is  an  arbitrary  constant.  (6)  |  sin  2  x  +  c,  |  sin  (3  x  —  7)  +  c, 

\  sin  x^  +  c,  in  which  c  is  any  constant. 


418  INFINITESIMAL  CALCULUS, 

Art.  43.     3.    Where  x  =  wir,  n  being  an  integer ;  where  a;  =  (4  w  —  1)  ^ 
±  .  485,  2  WTT  -  •  485.      5    -  -  cot  6>.      6.   cot  f;  60°.      7.    -2sin(2x  +  5), 


—  15  cos^  5  X  sin  5  a:,     2  aj  cos  a;  —  x^  sin  x,     -^^p-; ,     —  {m  cos  wa;  sin  mx 


2  sin  a; 
(1  +  cosa;)^ 

+  wcoswxsinwx),  e'^o«*(l  — a;sinx),  e^'^Cacoswix  — msin  wix).  8.  — cosx  +  c, 
—  2  cos  I X  -h  c,  —  i  cos  (3  X  —  2)  +  c,  —  ^  cos  (x^  +  4)  +  c  ;  c  being  an  arbi- 
trary constant. 

Art.  44.  3.  2  sec^  2  u  -  Du,  3  sec^  3  u  •  2)i<,  m  sec^  wi?< .  ?«',  2  w?(  sec^  wm^  .  u', 
2sec2  2x,  Jsec^ix,  wisec^wx,  6xsec2  3x2,  12x2sec2  4x3,  wmx'*-^  sec^  wix", 
6  tan  3  x  sec^  3  x,   12  tan^  4  x  sec^  4  x,   wm  tan«-i  9nx  sec^  mx,    |  tan  (|  x  +  3)  sec"-^ 

(|x  +  3),    orcosecx.     4.  tanx  +  c,    |tan2x  +  e,    |tan(3x  +  a)  +  c. 

sin  X 
6.  When  x  is  an  odd  multiple  of  -  and  dx  is  finite. 

Art.  48.  1.  -2csc2(2x+3),  isec(^x+3)  tan  (ix+3),  -3csc(3x-7) 
cot  (3  X- 7),  5sin(5x  +  2),  wsec^xtanx.  2.  -6cot  (3«  + 1)  csc2  (3«  + 1), 
secH|«- l)tan(^«- 1),  -  |csc2  2(i  +  5)  cot  f  (?  +  5),  -18  «csc2  9f-^, 
14(7  «  -  2)  sec  (7  t  -  2)2  tan  (7    -  2)2. 

Art.  49.     2.      ^^"-'  1  2  2 


Vl  -  x2«        Vl  -  2  X  -  x2        1  +  a^^        (1  -  x2)  Vl  -  5  x2 
^Vl  +  CSC  X.         4.   sin-i  x  +  a,      sin-i  x2  +  «, 


Vl  -  X2  VI  -  X2 

\  sin-1  x^  +  a,  in  which  «  is  an  arbitrary  constant, 
2  wx"~i  2  a 


Art.  50.     3. 


Art.  51.     1. 


X2«+1     '      1+X2'      V2«X-X2 

2  2        dy         2x         _3j^  ^         g  4 

1+4x2'     l  +  4y^dx'     l  +  x^'     1  +  y«  dx*         '1  +  16^2' 


4  «3  6  X      dx  K         2  1  -  x2  1  1 

—  •  0. 


1  +  «8'        1  +  9  X*  d«  '     1  +  X2'       1  +  3x2  +  X*'        Vn^'        ^(1  +  ^^)  ' 

^  „^^     .    7.  tan-ix  +  c,  tan-ix2  +  c,  itan-ix^  +  c. 

2(a  +  2x)Vx(a  +  x)      a^  +  a;2 

Art.  52.    2.^-^.     Art.  53.    2.        ^  "^  ^  "^ 


Art.  55.     1. 


x4-a*  Xy/x^-l      \/l-x2      Va2i:x2     a:Hl 

2 


l  +  x2 

Art._56.    2.  (3  x2?/2+3)  dy-\-  (2  xy8-|-2)dx,  3(y2_fl;a;)  ^y^_3(a;2_^y)  ^x,  etc. 

3    _J^       --^^        _(^Y(-Y~^      y  tan  X  4-  log  sin  y         ^     dx        dy 
^a;'  ^x'  \a)   \yl      '     log  cosx  -  xcoty'         '  2Vx     3Vy' 

i(^  +  ^V      ^(?!!:i^^  +  r^y     ,(ytanx  +  logsint,)dx-(logcosx 
•^  ^  V  X      Vy  '  \    a"»  0"*    / 

—  X  cot  ?/)  dy. 


ANSWERS,  419 

Page  82.    1.  (i)  24  x^  +  15x^  +  124 x  +  55,  (ii)  a-\-b-[-2x,  (iii)  (a  +  x)'«-i 
(6  +  a;)-'  [m(6  +  x)  +  «(«  +  x)],     (iv)  ('"^  -  "^  +  "'6  -  »«) (x  +  a)-i ^ 

(X  +0)"+' 

(V)  (rn  +  mx  -  nx)xr^-^  .     a^ .      1 

(yiii)  ^(^-^) -,     (ix)   -l(l+        ^       V      (X)  _i^L_, 

2v^A/a  +  a;(Va  +  Vx)2  a;3V        Vl  -  x*/  xVl  -  ^2 

aHa%^-4r^     ^^  28x3  +  6x-17  j     -2a2^  "«      , 

Va2i:^2  ^^  7a:*+3x2-17a;  +  2^       a*-x*^       xVa^^^ 

(iv)secx,  (v) 3.  (i)20x*cos4x^  (ii) -7sinl4x,  (iii)  6sec2xtanx, 

vT+x2 
(iv)  8sec2(8x  +  5),       (v)  x'»-i(l  +  wlogaj),       (^vi)  pqx^'^sinp-^x^cosx^, 

/■  ••\     /  •      N,.  1   •    /     .  1N         /  "-N         /-  •      \                  /•  ^  cos  (log  nx) 
(vii)  7i(sinx)"-^sm(n  +  l)x,      (viii)  cos  (sin  x)  •  cos  x,      (ix)  ^^— ^^ — ^, 

(x)  7icotnx.       4.  (i)  — i— ,  (ii)  ^ ,   (iii)  — —       6.  (i) , 

^  ^  ^      X*  +  1  ^   ^  tan2x  _  1 '   ^    ^  1  _  x4  ^  ^  e^+e-^ 

(ii)   _i,  (iii)       -^     ,         (iv)    ?i ,  (V)    -^^'-^', 

y/i  _  a;2  cos2  X  +  ;i  sin2  x  a  -\-  b  cos  x 

2     I 

(vi)  e"*  sin"*-!  rx(<:<  sin  rx  +  ?«r  cos  ?'x),  (vii)   — ^ — —  log  a  •  a^, 

(iii)x«v!-t^^^^,  (iv)  e*V(l  +  logx),  (v)x(-').x-{i  +  logx+(logx)2|, 

Cvi^  x=«'+ia  +  21oex^        8  m       ^"^  +  ^y  +  ^       m      a;    2(x2  +  y2)  _  ^2 

(VI)  X       (l  +  21ogx).       8.(1)       /^^  +  ^,y^^,      00       y-2(x2  4-l/2)  +  a2' 

.....  2xy*  ,•  N  If  ,     N       1     /  N         cos X (cos  ?/ +  sin  ?/) 

(ill)  —  ,    .,  a  ■ ?    (iv)  -{msec  (xw)  — w},    (v)  — -. ^ — -. r^^, 

^    ^      4:  x-y^  ■\- cos  y     ^    ^  x^  \  if/     i^^^    \  y      sin  x (cos  y  —  sin  y)  —  1 

^"^'^  eJ'  +  x'        ^    ^  x2  -  x?/  log  x'        ^     ^  x(l  +  »iy)  ^-  (1  +  log  x)2 

10.  (i)2y-|,      (ii)8«-ll,      (iii)  sec  x,      (iv)  -  cot  2;,      (v) ■ 

11.  (i)  (12  x3  +  18  X  +  5)  (6  x2  +  3),  (ii)  (e**" «  +  2  tan  t)  sec2 1,  (iii)  g, 
(iv)  f^P^-  12-  (i)  90%  (ii)  73°41'.2,  (iii)  90°,  (iv)  2°21'.7,  (v)  70''31'.7. 
14.  Speed  of  Q  in  inches  per  second  is  116.82,  225,  7,  319.18,  390.9,  436, 
451.39,  390.9,  225.7,  respectively. 

CHAPTER  V. 

Art.  59.     1.   The  lengths  of  the  subnormal,  subtangent,  tangent,  and 
normal,  are  respectively  :  (1)  3,  5^,  6|,  5  ;  (2)  4,  4,  5.66,  5.66  ;  (3)  -  ^, 

-  ^ln^,  _L  V'(a2-Xi2Xa*-c2xi2),  ^^^^~^^^i^  e  being  the  eccentricity  ; 
Xi        axi  a2 

(4)  sin  xi  cos  xi,  tan  xi,  tan  Xi  Vl  4-  cos2  xi,  sin  xi  Vl  +  cos2  xi ;  (5)  yi^,  1, 


420  INFINITESIMAL   CALCULUS. 


vT+y?,  yi  VI  +  yi^.     2.  Where  x  is  infinitely  great.     3.    Infinitely  great. 

-----  -i  -h  i  0  0 

6.   xxi  ^  +  yyi  ^  =  a^.    7.  xxi  ^  +  yy\  ^  =  a^     8.   a  sin  6,  2  a  sin^  -  tan  -, 

2  «  sin  -,   2  a  sin  -  tan  -•     12.    90°,  0°,  cot-i  4^,  i.e.  32°  12'. 5. 

2'  2         2 

Art.  61.     1.  (1)  a,  a^^  a  Vl  +  ^'^  rVIT^;  (2)  — ,  2  rd,  -^4:6  +  0-\ 

2r  2 

aV^OT+T^;  (3)    -^,   -a,  -^/a^~+l^,  _VaM^;  (4)  ««^'-i,  ^^^, 
a  a  n 

a^"-i  Vw2  +  d^,  -  VwM^.      3.  ar,  -,  r  V'l  +  a^  -  VHTo"^.      4.   (a)  lA  = 
n  a  a 

34°  65'. 2,    0=:74°55'.2;    ^p  =  50°4V.9,    120°41'.9;  (6)    \i/ =  26°33'.9, 

4>  =  55°  12'.8. 

Art.  62.     1.    In  feet  per  second:  0,  4;  2.828,  2.828;  3.57,  1.79;  3.77, 

1.33.       Solution  for  x  =  2:   Where  x  =  2,  the  tangent  to  the  parabola  has 

a  slope  1.     Accordingly,  the  moving  point  is  there  going  in  a  direction 
which  is  at  angle  45°  to  the  ic-axis.     Hence,  the  speed  of  the  x-coordinate 

(i,e.  ^^  =  ^  X  cos  45°  =  4  X  —  ;  also  ^  =  4  x  — .1     2.  20  and  22.36  ft. 
V        dt)      dt  V2  <^i  V2  J 

per  sec.         Suggestion :     Differentiation  with  respect  to  the   time   gives 

2  y^  =  4— .1     3.    .399  and  -  9.97  ft.  per  sec. ;  9.7  and  -  2.425  ft.  per  sec. 

4.  442.82  and  161.6  ft.  per  sec.  ;  199.15  and  427.08  ft.  per  sec.     6.   (1)  (2,  8), 
(-  2,  -  8)  ;   (2)   a,  ^h),  (-  h  -  3rh)  ;  (3)  300. 

Art.  64.  2.  ^,  h.  3.  The  tangent  at  the  middle  point  of  a  parabolic  arc 
is  parallel  to  the  chord  of  the  arc.  4.  —  S  ±  V^ ;  find  the  abscissa  of  the 
point  where  the  tangent  is  parallel  to  the  chord  joining  the  points  whose 
abscissas  are  3  and  4. 

Art.  65.  3.  25.1  cu.  in. ;  ^i^.  4.  4  Trr^  •  Ar;  50.3  sq.  in.,  502.7  cu.  in.; 
^h^  ^h^  ^-     5.   1.35  sq.  in.  ;  7:5  approximately. 

Art.  66.      2.   (1)    _  1,    -  1,  I ;      (2)    -  f,   -  I,   _  1  ;      (3)  2,  2,  3,  4  ; 

(4)  -  ^,  -  I,  3,  -  I  ;  (5)  2,  2,  -  3,  -  3,  1.     3.  n-r-^  =  Ap»(n  -  2)«-2. 

Art.  67.     4.   1.6,  .4.     6.   Jr2,  i.e.  2  6'^-  .0048,  .035.    7.   .0349,  0,  .0025. 

9.  J  «±^,  J-«+^  ;  ^.         10.    2.41,  .1.         11.  a  Vrn"2,  1  y/WT^, 
'     X        '     a         ^x  a 

12.  .078.     14.  TTX^  Tzx^.     15.  5.03,  10.05.      18.  10.37,  5.06.      19.  J^^^^=-^, 

^  a^  —  x^ 

^Va2-x2,  ^^'^(^^-^'^  ^^Va2_e2x2,  e  being  the  eccentricity.    20.  ^, 

—  —2  /I  « 


a  a 

a,  r  cosec  a,  V2ar. 


ANSWERS.  421 


CHAPTER  VI. 


Art.  68.     1.  (i)  -—^^  ;       (ii)   8  +  ^  +  -1=  ;        (iii)         ^°^  ^ 


(l  +  x--2)2'       ^'  0!^     4Vx^'  ^   (l-sinx)2' 


(«2  +  ^2)2'  sin^X  (1_  3,2)1 

24(l-10x2  +  5x^).      4    ^.     _4!  _8e^sinx.       6.  (i)   -^; 

^  ^  (1  +  a;2)5  ^  ^       x2  '  ^  ^  ^  ^       aV 

^^   -  n  T9.V0     ^-  ^^  -^•^'  -^•^^>  <^"^  ^'  -^20.    9. ^;   -i- 

i^+^yy  4asin4^ 

,  2 

12.  24  ic.     13.  ^  =  I  x2  +  2  X  +  Ci,  y  =  I  x3  +  x^  +  cix  +  C2,  in  which  Ci  and 

dx 
C2  are  arbitrary  constants.         14.    Sy  =  x^  —  9 x  +  19.         15.    y  =  4 x^  4-  x. 
16.  (2)  '—  f  ft.  per  sec'  per  sec.    17.  In  '  in.  per  sec'  per  sec  :  (i)  1152  tt^, 
(ii)  768  7r2,  (iii)  384 7r2,  (iv)  0.        18.  s  =  igt^  +  Cit  +  C2.        19.  15.5  sec, 
3881.9  ft.    20.  ^Vi)Sec 

Art.  69.     2.   e^  a^(loga)»»,  a''e«^  ft^a*^  (loga)«.  4.   cos^a;  +  — V 

«    •    /       ■  W7r\       „        /       ,  7i7r\        -     (-l)»-i(>i-l)!     (-1)«-I2.(7i-1)! 

«» sin  ax 4- —  ,  a»cos  axH )•     5.  ^^ ^ ^^ —■,  ^ ^ ^^ ^• 

V  2  y  V  2  /  x«  (x-2)» 

g    (-  l)»;t!^  (-  \Yn\  ^  2.n!  (-  l)^ac''(w  +  n-\)\ 

x«+i     '  (1  +  a:)"^i'  (3  -  x)«+i'  (m  -  1) !  (6  +  cx)'«+«  ' 

l(l  +  x)'»+i      (l-x)«+i/  l(l-x)«+i      (l+x)~+i/ 

Art   71      2    ^_±A22^,    _  ^  +  «  cos  ^ 
6  sin  0  ?)2  sjn^  ^ 

Art.  72.     2.  (x4-120x2  +  120)xsinx-20(a;2-12)x2cosx.    3.  (x  +  n)e*, 
2«-i(Ai  +  2x)e2*. 

Art.  73.     3.    (1)  y'  =  xy"  ;    (2)  x^y"  +  2y  =  2xy';    (3)  y'  +  2  xy"  =  0 

(4)  (ix^-2y^)y^' -ixyy' -x'^  =  0;  (5)  yy'  =  x(yy"  +  y^-').  4.  (1)  y' =  0 
(2)  y=x?/';  (S)y"  =  0;  (4)  y"  =  y  ;  (5)  y"  =  m'^y  ;  (i6)y"  +  m^y  =  0 
(7)  y"  +  wi22/  =  0.    5.    ^2(1  +  2/2')  =  r2  ;  x2(l  +  y2')  =  r-f'' ',  {1  +  J/^'f  =  ry". 

CHAPTER   VII. 

Art.  76.     4.    A  minimum  ;  neither  a  maximum  nor  a  minimum.     8.    See 
Ex.  3.     12.  See  Ex.  3.     13.   (l)_Min.  f or  x  =  ^  ;  max.  for  x  =  -  2.     (2)  Min. 

at  -'^-^^ ;  max.  at  i:l±y^.     (3)  Max.  for  x=0  ;  min.  for  x  =  ll^^; 
6  6  ^  ^  12 

min.  for  x  =  -^ — '— ;  for  x  =  2,  neither  a  max.  nor  a  min.     (4)  Max.  for 

12       '  ^  ^ 

X  =  —  1 ;  min.  for  x  =  I ;  neither  a  max.  nor  a  min.  for  x  =  2.  (5)  Min.  for 
X  =  4.  (6)  Max.  when  x  =  —  4,  and  when  x  =  3  ;  min.  when  x  =  —  3,  and 
when  X  =  4.     (7)  Min.   for  x  =  16  ;    max.  for  x  =  4  ;   neither  for  x  =  10. 


422  INFINITESIMAL    CALCULUS. 

(8)  Max.   for  a;  =  —  10  ;    min.   for  x=—2;    neither  for  x  =  2.      (9)  Min. 

value  is  — ,  i.e.   —.3678.     (10)  Max.  when  x  =  e.     (11)  Max.  value  =  8  ; 
e 

min.  value  =  2.     (12)  Max.  or  min.  when  sin  a;  =  Vf  according  as  the  angle 

X  is  in  the  first   or  the  second   quadrant.       (13)    Max.    when  ic  =  cotx. 

16.  (av'2,  a ^4). 

Art.  77.     7.   Each  factor  =  Vthe  number.  8.   -•  9.   A  square. 

10.    (i)  (a^  +  b^)^;   (ii)  a  +  2Vab  +  b;   (iii)  2ab.      11.   Let  the  perpen- 
diculars drawn  from  A  and  B  to  MN  meet  3IN  in  B  and  ^S*  respectively  ; 

then  (1)  BG  =  CS;  (2)  BC  =  ^^^^^     12.  (i)  f  r;  (ii)  f  r.     13.  19°28'. 

14.  (i)  Vol.  =  .5773  vol.  of  sphere;  (ii)  height =rV2.     15.   (i)  Vol  =^\Tr  a^b  ; 
(ii)  height  =  ^  &.       16.    1.       17.    2s  le.  114°35' 29".6.      18.    fa.      19.    1:2. 

22.    1|  times  the  rate  of  the  current.        23.  —  d,  ^  d.       24.    (^J  +  6^)i 


(1)   (0,  0)  ;      (2)   (3,   -  3)  ;      (3)   (f ,  W)  I      (4)   (2,  f ) ; 
where  x  =  0,  and  where  x  =±  V3  ;    (7)  where  x  =  0, 


25.     «  . 

V2 

Art.   78. 

1 

3^ 

2, 

);  (6) 


and  where  x  =  ±  2  \/3.     2.   (1)  Where  x  =  —  ;  (2)  where  x  =  — ;  (3)  where 

5  4 

x  =  ±-^;    (4)   (c,  &);    (5)   (cm);    (6)    (^^,  ^)- 

CHAPTER   VIIL 

Art.  79.     2.   3  x2  +  e'  sin  y,    4  ?/  +  e*  cos  y  —  cos  2  sin  ?/,    6  2  —  sin  z  cos  y. 
3.    (a)  -:^  and  -lli|, ;     (6)  -Zli^,  and  -^^4  .     ^^^  ^20  ^^^    -4|^ 
4V1I9  5VII9  3V'89  5\/89  3V47  4>/37 

respectively. 

Art.  81.     3.    Increasing  3^::  units  per  second.    4.    Decreasing ~ 

units  per  second.  20  V 119  5V89 

Art.  82.     3.  .036;  .036011.    4.  (i)  ^<^y-y^^^.  (ii)  y-\ogy  .dx-hxy^-Ulf/; 

y         ^  +y  /logy        logx 

(iii)  yxv-^  dx  +  xv  log  x-dy;  (iv)  -  dx-\- logx- dy,  (v)  it  I  — ^  <^^  +  — rp  <^2/ 
5.  .025.    6.  2.2;  2.37.     7.  .0017.    8.  xy''-'^(yzdx+zx\ogx  -  dy+xy\ogx  -  dz). 
Art.  83.     3.  4.72  sq.  in. 

CHAPTER   IX. 

Art.  90.      /sf^V-^^Uf^'' 
I    [dyV       dydy^i      \dy, 

—  4asin  -. 
2 
a.     4.   -  (a2  sin2  e  -\- b"^  cos^  $)  2  ^  a6. 


ANSWEBS.  423 

Page  147.  1.  (i)  ^i^-2y^  =  0.  (ii)  ^  +  ^  =  0.  2.  ||-2f^V 
=  cos2  ^.      3.    (i)  ^  +  M  =  0.      (ii)  ^  +  ?/  =  0.       (iii)  ^  =  0.       (iv)  ^ 

+  a2y  =  0.        (V)   g+y  =  0.        (vi)   |^+2g+2/  =  0.        4.    (i)tan«; 

(ii)   —  3  sin*  fcosi ;  3  sin^  i(4  —  5  sin^O. 

atcos^t 

CHAPTER  X. 

Art.  97.  3.  y  =  x^  ;  y  =  x^  -  ?A7  ;  y  =  x^  +  514  ;  y  -  k  =  x^  -  h\ 
4.  ?/=:4x  +  c;?/  =  4x;«/  =  4ic  —  5;y=:4x  +  20.  5.  y  =  4  x^  +  c  ;  y  =  4  x'^ ; 
?/  =  4  x2  -  2  ;  ?/  =  4  x2  -  13  ;  y  =  4  X  -  62.  8.  16  «2  ;  64  ;  256  ;  400  ;  16  «2+ 10, 
etc.  ;  16  «2  4-  20. 

Art.  98.     3.    |.     4.    2 ;  0.     5.   4 ;  0. 

Art.  100.  4.  (a)  2  y  =  x'^,6  y  =  x^,24  y  =  X*  ;  (b)  y  =  x""- ■]- 5x,6y -2  x^ 
+  15x2,  12y  =  a;*  + lOx^;  (^^  y  =  1 —cosx^  y  =  x  —  s'mx,  2y  =  x--\-2cosx—2  ; 
(d)  y  =  e'  -\,  y  =  e""  -  X  -  1,  2y  =  2  e'^  -  x"^ -2x -2.  6.  y  =  1,  y  =  2, 
y  =  cos  X,  y  =  e*. 

CHAPTER   XI. 

Art.  104.     9.    I  x»  +  c,  /^  »^'^  +  c,  tr  ^"  +  c,  -  |  x-is  +  c,  -  ^^^  x-i^  +  c, 

-r^,+  c,  -4  +  <'.  f^^  +  c,  —i — x'^^-^^  +  c,  |x^  +  c,  ^x^+c,  8Vx  +  c, 
2  x^  X*  \/2  +  1 

-  — +  c,    -_§-  +  c.  10.    iv^  +  c,  ^M^  +  c,   ^+c,  12s^+c. 
v9.               14  X*  2^^4        . 

'+» 

m+n  m+S  6+n  

11.    _£ZL_x"^+c,  -l^t~^  +  k,  -J^v^  +  c,  ?:^^^+c.     12.  logc^, 

m  +  >i  m  +  3  6  +  w  t  -\-  s 

logc(s  +  2)2,  -  ilogc(7 -a^),  logc(4«2_3^  +  ll).     13.    e« -j.  c,  | e^  +  c, 

2e^'  +  c,  ^ h  c,  — ^^^ f-c.     14.  -  icos3x  +  c,  isin7x+c,  nan5x+c, 

log  4  2  log  10 

-  cos  (x  +  «)  +  c,  ^  sin  (2x4-  «)  +  c,  f  tan  ( -^  +  -  j  4-  c.     15.    ^  sec  2  x  +  c, 


fsecf  x+c,  sin-i^  +  r,  ^sin-ix24-c,  |sin-i5x4-c,  f  sin'^x^+c,  log(l  + vl+«2) 
+  c,  \  tan-i  «2  _|.  c^  tan-i  2  x  +  c,  sec-^  t  +  c,  sec-i  3  x  4-  c,  i  sec-i  x2  4-  c, 
I  vers-i  3  X  4-  c  or  ^  sin-i  (3  x  —  1)  4-  c,  \  vers-i  4  x  +  c  or  ^  sin-i  (4  x  —  1)  +  c. 

16.   ^  -  f  «3  4-  16  <  4-  c,      a^x  +  V-  «^x5  4-  f  a^x^  +  j\  x''^'  +  c,      -e"""  +  c, 

1.^1 

-  sm  «x cos  nx  +  c. 

[In  the  following  integrals  the  arbitrary  constant  of  integration  is  omitted.] 
Art.  105.     11.    I  sine  ^^  ^^I^  (3  +  2  tan2  x),   -  ^  tan  (4  -  7  x),  -  ^  e-*'. 

12.   log(x  +  l)4-,f^^+f^^,   f +3x+31ogx-l,  f(x4-2)^(x-8),^5(x-2)^ 


424  INFINITESIMAL   CALCULUS. 

(2^  +  3).  13.    f(x+a)3,    Hm  +  nx)\    _  3  Vs-^y^,   i(44-5i/)l 

14.    Igm+nx^   -■^r~l->  log(tan-ix),  -COS (log X).     15.  r'5(«-l)2(3«  +  2), 

35  1 

—  («  +  &2/)^,  {(w  +  0)*,  f  sin  I  a:.    16.  |sinx(3-sin2x),  ^tanx(tan2x+3), 

I  cos3  a;  -  cos  X  -  i  cos^  x,  n  tan  ( -  J  •  17.   -  -^  log  (3  +  7  cos  x), 

-^log(9-2sinx),   _  |  V4  -  3  tan  x,  _Lsin-i /^^^^^l^V     ig.    Va2  +  x2, 

Va2  _  x2  X 

Art.  106.     7.  ^(ax-1).     8.   -(x  +  l)e-^.     9.  ae^(x2  -  2  ax  +  2  ^2). 

10.   xlogx-x.     11.    ^x2(logx-^).      12.    ^x3(31ogx-l).      13.  xtan-ix 

-  log  vTTx2.  14.  i  (1  +  x2)  tan-1  x  -  |  x.  15.  2  cos  x  +  2  x  sin  x  -  x-  cos  x. 
16.  e^[x'»  -  mx"*-!  +  m{m  -  l)x'«-2  _...  +  (_  \)^-'^m{m  -  1)  ...  3  •  2  •  x 
+  (-l)'».w!].     17.   -  ^xco6  2x  +  I  sin2x.      18.   —  Vl  -  x^  .  sin-i x  +  x. 

Art.  107.     7.   ^-  tan-i  ^-±^ ;  sin"!  ^^ ;  log  (x  +  3  +  Vx-^  +  6x+10) . 

2V2  2V2  V26 

8.    ilog^-±^^;    sin-i^^±^;    log(2  x  -  5  +  2  Vx2  -  5  x  +  7).  9.   -^ 

1-^  _  V53  V33 

^^^2x  +  5-V33.    _^iog2^±A.:-^;   :iog(8x-3  +  4V4^233^T5). 
2  X  +  5  +  V33      Vei        2  X  +  5  +  V61 

10.  ^tan-i^^^;  lsin-i^^±^;  _4^  ^^g  Vl37  +  5  +  8  x. 
\/71              V71                                 13  V137        V137-5-8X 

11.  vers-i?  and  sin-i  ^^=i ;    1  vers-i  —  and   \  sin-i  ^  ^  -  ^  ■     1  sec"!^. 

4  4^  9  ^  9^^  5 

12.  isec-i^^;  i(xV9 -x2  + 9sin-i^V  ^-i-ir.  13.  xV9^^  +  9sin-i?; 
logtan^^  +  ^V  1  log  tan i^^.    14.  |logsec(3x+«);  1  logsin(4x2+«2). 

|logtanfx  +  ^V      16.   -^2^:11^;   — ^=5  -^51ZZZ. 
^     °        \        4/  75x8      '  4V4T^'  6x 

Art.  108.     3.    log  (X  +  3)2  +  — i_.  5.    log  (x2  +  4)2  (x  -  l)^ ; 

^^-^(f^T-^""1'  '•  '^^Iri?'  '•  l-«^(2x-f5)(x-7)a. 
8.   ^x2-2x  +  log('^  +  ^)'.       9.    ix2  +  log:^^^^^.       10.   log ^ —  + 

X  -  1  X  (X  —  1)2 

|log(2x+5).     11.  log^^=£K^±^.     12.  log(x-3)2(x+3)8(x-2)(x+2)6. 

X 

13.  log  (X  -  1)  -  -^.         14.  log  V4¥+^  +  ^         .         16.  log  X  + 

X  -  1  4  (4  X  +  5) 


ANSWEB8,  425 


flog(2a;+5)  +  --     16.  log  (x  +  4y  V3  x  +  2  + -—f——-     17.  log(x4-l)2 
X  3(3x  +  2) 

+  ^"^^t-      18-  log  "^  -  ^^  ^^1^"' -V-      19.  I  log  (3  X  -  2)  -  i  log  (x2  +  5) 

__Ltan-i— .    20.  log X  +  2  tan-i  X.     21.  x  +  ilog^^-±-^-\/3  tan-i— . 

V5  V5  ^  x2  V3 

22.  log x2  +  \/3  tan-i -^.      23.  log x3(x2  +  3)2.      24.  2  logx --- 2  tan"*?. 

V'3  ^  ^       2 

25.  log ^^^^ +  i  tan-i  ^^:^-      26.  tan"!  x  +  log  VxM^  - 


x2  _  2  X  +  5      "  2  "  x2  +  1 

Art.  109.     4.   e^  cos  y  ;  x^  4-  4  x2?/  +  4  x  —  6  y.         5.   cos  x  tan  ?/  —  sin  x  ; 

?y2 
xey  —  2  xy  +  x2 ;  3  x  -  2  x2  -  xy  —  ^• 

2 

Page  190.     I,  JIL^ +  c,       ^x2(«+6)  +  c,       -J-±l—zn+t+s^c, 

V2  +  W  +  1  w  +  «  +  3 

-f-2/'-«  +  c,   -12x\  +  291ogf,   ^H-8?;-flog(«2  +  3)-llV3tan-i-^  +  c, 
rts^  ^  ^  V3 

^-  2x  +  f  log  (x2  -  2)  -  -A^  log  ^z::^+  c,  -1^  tan-i  ^4-  c, 

2  2\/2x  +  v^  6V5  2V5 

_JL  log  ^  ~  ^  ^  +  c,        7  a^  +  H«- a^  +  H-1-,        ^sin-i-  +  c,  i  log  (x^ + 

4V3        Z  +  2VS  3 

Vx6  -  9)  +  c,     i2+— — ^— -+ilog(2  2;-l) +c.  2.    -log sec  (wx 

8  (2  2:  —  1)  -   m 

+  n)  +  c,        ^  tan  3  X  +  t  log  (sec  3  x  +  tan  3  x)  +  4  x  +  c,        00,        2.4288. 

3.  X  cos-i  X  —  Vl  —  x2  +  c,        X  sec-i  x  —  log  (x  +  Vx'^  —  1)  4-  c,        x  cot"i  x 

X 

4-  i  log  (1  4-  a;2)  +  c,  x{(log  x)2  _  2  logx  +  2}  4-  c,  -  «e'«(x2  +  2  ax  +  2  ^2)  4.  c, 

-  (x^  4-  3  x2  4-  6  X  4-  6)e-*  4-  c,  cos  x  (1  -  log  cos  x)  4-  c,  -^^^ log  x ^- 

m  +  lV         1^4-1 

4-c.        4.    fx^-fVx  +  c,   18(|x^4-ia;^  +  ix^4-a;'b +91og^    ~ '^  4- c, 

11  qi      1        , , X*  4- 1 

4  (3*  -  2*)  4-  4  log  ^-— ^,    \/x2^n:  4-  log  (x  4-  Vx2^^)  +  c.      5.  .206  (the 

2^-1 


base  being  10),  1  ^  -  1 V  \{f-V),  -^\%ir^    6.  -  ^^  log  (m  4- n  cos  0)  +  c, 

log  (sin  e tan ^U  c,  J-  log  tan  f ^  4-  ^U  c, ^-^ ^^  log"^^^"^ 

V  ^/  V2  V2      8/  8(sec2x-4)      ^^     ^secx4-2 

+  c  (see  result  in  Ex.  3,  Art.  118),  sin-i  /l^IL^A  4.  c,  —  log^  (ms  +  w)  +  c, 

V    2    /  3m 

_l-sec->«!+c,    tan-ie.  +  c,     }log?i^+c,     J  log  I+i5!t|»  +  e, 
»n  log  a  m  e*  +  e-==  1  —  tan  2  ^ 

4  \/2  sin-i  (  V2  sin  -  j  4-  c,  cos  x  cos  y  —  ?/2  _|_  ^^  _)_  c^  cos  x  sin  y  +  x  —  y  4-  c. 


426  INFINITESIMAL   CALCULUS. 


CHAPTER  XII. 

Art.  111.  5.  (6)  76.  6.  18.  8.  5.  11.  -2/  VS.  13.  (a)  2  ;  {d)  4. 
16.  .862025;    6.644025;    .862;    .401.  17.   (1)  -^ ;    (2)  lOf ;    (3)  3.2; 

(4)  68^*3  ;  (5)  I  a2  .  (6)  12  V2  ;  (7)  No  area  is  bounded  ;  (8)  (a)  log  7,  i.e. 

1.946  ;  log  15,  i.e.  2.708  ;  log  n  ;  A;2  log  ^       18.  -W  Vf . 

a 

Art.  112.  9.  -W^TT.  10.  iQj^iTT.  11.  ^^'w.  12.  (a)  f(2v^-l)7r; 
(6)  K4  ^2  -  l)7r.     13.  -V^3.  ^.     18.  405  (|  - 1)  ,r,  225  ^|  -  ^)  tt. 

Art.  113.  2.  ?/2  =  48  x  -  80  ;  24.  3.  sc  -  4  =  2  log  ?/.  4.  cc  -  4  =  4  log  ?/ ; 
4.  5.  3  y2  =  16  X.  6.  5  1/2  =  48  x2  -  112  ;  the  conies  if  =  kx^  -}-  c,  k  and  c 
denoting  arbitrary  constants.  7.  3  y  =  a;2  +  6  ;  the  parabolas  y  =  kx^  +  c, 
k  and  c  being  arbitrary  constants.  8.   ?/2  t=  7  x  +  4  ;    the  parabolas 

1/2  z=  A:x  +  c,    A;  and  c  being  any  constants.  9.    The  circles  r  =  c  sin  ^  ; 

r  =  4  sin  d.     10.  r^  =  ce^  ;  r2  =  4  e^.     11.  r  =  a(l  —  cos  ^),  in  which  a  is  an 
arbitrary  constant. 

CHAPTER   XIII. 

Art.  116.     1.  f  v^ic  (  Vx  -  3)  +  4  tan-i  ^x  +  c.     2.  2(  Vx  -  tan"-iv^)  +c. 

3.    i(3x-2)^-^-A___+c.  4.    ^2^(2  +  x)^(5x+17)+c. 

3  v/3  X  -  2 


6.   -  I  log  (7  +  5  \/2  -  X)  +  c.     6.  X  +  1  +  4  Vx  +  1  +  4  log  (Vx  +  1-  1)  +  c. 
Art.  117.     5.  ^\/4x2  +  6x+ll  +  | log  (2  x  +  3  +  \/4x2  +  6x+  11)  +  c. 

6.   -3Vl2-4x-x2-10sin-i^±^  +  c.     7.  JL  ipg^gEg^"  v^^+^4- c. 

4  2V3        \/6-8x  +  V6  +  x 


8.  3sin-i^+g-^log^^-^^-^^J:;+c.       9.  log^-^+ ^^'^+^+1  + c. 

4         \/3        V6-3X+ Ve+x  x+l+Vx2+x+l 

10.  Vx2  +  x  +  l4-ilog(x  +  ^  +  Vx2  +  x  +  l)-31og^~"^+^'^'  +  ^+^  +  c. 

,,       ,  1X  +  2    ,  X  +  1  +  \/x2  +  X  +  1 

11.  isec-1— ^^^ — ^f.^ 

4 

Art.  120.  2.  ^  cos^  x  —  cos  x  +  c ;  sin  x  —  |  sin^  x  +  c  ;  f  cos^  a;  —  ^  cos^  x 
-cosx+c.  4.  (1)  |cosix(cos2x-4)+c;  (2)  5sin^x(|-^sin2x+5Vsin*x)+c; 
(3)  2Vsinx(  1  -  I  sin2  x  +  f  sin*  x)  +  c;  (4)  3  cost  jc  (^J^  cos2  x  -  ^)  -\-  c. 
7.(l)^tan8x+tanx  +  c;  (2)-^cot8x-cotx+c;  (3)^tan6x+|tan8x+tanx+c. 

9.  (1)  j\  tanS  X  (3  tan2  x  +  5)  +  c  ;  (2)2  tant  x  (|  +  ^  tan2  x  +  A  tan*  x)  +  c ; 
(3)  f  tanf  X  (^  +  I  tan2  x)  +  c  ;  (4)  sec^  x(^  sec*  x  -  ^  sec2  x  +  i)  +  c ; 
(5)  ^  Vcsc  x(6  -  csc2  x)  +  c ;     (6)  -  esc*  xQ  esc*  x  -  §  csc2  x  +  |)  +  c. 


ANSWERS.  427 

Art.  121.     3.  (l)\(^-sm2x-\-^^^^\+c;       (2)  ^JgCSx  +  4  sin2  a; 

-ismB2x  +  fsm4x)  +  c;  (3)  ^  _  ^j^^jl  _  sm^  2  x 

^  *  ^         '  ^  ^  16         64  48  ' 

(4)  ^V  cos  2  x(cos2  2  X  -  3)  +  c  ;  (5)  ^l^f 3  x  -  sin  4  x  +  ^^^^\  +  c. 

Art.  122.     1.  (1)  _s^na^cosx_^g_^^.  ^g)  _  ^sin2xcosx  -  |  cosx  + c  ; 

(3)  -^^^^-5^(sin2x+f)  +  fx+c;  (4)  -lsin4xcosx-i^^^(sin2x+2)+c. 
4  15 

2.  (1) -cotx  +  c;  (2)  logtan--|cotxcscx-fc;  (3) -i-^^ -|cotx  + c. 

2  sin^  X 

5.  (1)  ^sinxcosx(2cos2x+3)+f  x  +  c;   (2)  ^  sin  x(cos*x  +  f  cos2x+|)4-c  ; 

(3)4  ^^^^  4-|tanx+c;  (4)  ^tanxsec^x+f  secxtanx+f  log(secx+tanx)  +  c. 
cos^  X  I         v\ 

6.  (1)  I  tan  X  sec  X  +  ^  log  tan  f  J  +  -  J  +  c  ;  (2)  i  tan  x  (sec2x  +  2)  +  c  ; 

(3)  i  tan  X sec^x  +  f  |  tan  x  sec  x  +  log  tan  (7  +  -)  }  +  c.     7.  (1)  ^  log  tan  - 

—  I  cot  X  cosec  X  +  c  ;    (2)  —  ^  cot  x  (cosec2  ^  +  2)  +  c  ;    (3)  —  |  cot  x  cosec^  x 

—  I  (  cot  X  cosec  X  —  log  tan  -  J  +  c.  11.  (1)  |  tan^  x  —  log  sec  x  +  c  ; 

(2)  — icot^x+cotx+x+c;  (3)  ^  tan^x— tan  x+x+c;   (4)  ^  tan^cc— ^  tan2a; 
4-logsecx+c.     14.  (1)  |(sinxcosx+x)  — ^sinxcos^x+c;  (2)  —  ^sinxcos^x 

+  2V  sin  X  cos3  x+  x^5  sin  x  cos  x  +  j^^ x  +  c  ;  (3)  -  i  ^^  (3  -  cos^x)  -  —  +  c. 

2  sm  X  2 

17.  (1)  -lcot7x-|cot5x+c;  (2)  ^tan^x+c;  (3)  -TVcot3x(3cot2x+5)  +  c. 

Page    222.       3.    (1)    3  «i  +  ^  log  ^^i^^  -  V3  tan-i  f^-^l+^W  c  ; 

t  -  1  V V^     / 

(2)3(2.  +  3)^^.  (3)^tan-i(^^^Uc;  (4)-I- log^^EM^VS  ^^. 
8\^7+l  V5  \Vl-4x2/  2\/5       Vl-4x2+\/5 


2  V4x— x2    _i  X 


(5^  _£.^15±II:L_vers-i-+c;  (6)2Vx2+3x+5-21og(x+f +Vx2+3x  +  5)+c; 

(7)21og(x  +  |+Vx2+'3x  +  5)  +  -Llog^Q  +  3^-^^^(^^  +  '^^  +  ^)  +  c; 
V5  aj 

^^~Vi     ^ ^T^: ^^^~^^n(x2-16)2     32(x2-16) 

^.t.-.:4^n-^  C-)iS^2-^_^t.an-.|.c. 

^^^^-^^-i^)^-     (12)cos-.(^)-2Ve^,,. 


CHAPTER   XIV. 

Art.  125.  2.  2525.   3.  3690  ;  3660  ;  (true  value  =  3660).   6.  333  in 
20,000.   7.  .05075;  1509. 


428  INFINITESIMAL   CALCULUS, 


CHAPTER   XV. 

Art.  129.  4.  The  parabolas  ?/  =  3  x^  +  cix  +  Cs,  whose  axes  are  parallel 
to  the  ?/-axis  ;  2 «/  =  6  x2  +  11  x  -  13  ;  ?/  =  3  x'^  +  15  x  +  22.  5.  The  cubical 
parabolas  y  =  x^  +  cix  +  C2  ;  y  =  x^  -\-  x;  y  =  x^  —  x  +  i.  6.  The  cubical 
parabolas  y  =  cx^  +  CiX  +  C2,  in  which  c,  Ci,  c^  are  arbitrary  constants ; 
6  ?/  =  x^  +  11 X  ;  5 1/  +  x^  +  16  =  22  X.  7.  The  cubical  parabolas  x  =  Ciy^ 
+  C22/  +  C3;  120x  =  lly3-251?/  +  240;  7x  +  4?/3  =  622/ -  85.  8.  15,528  ft.; 
62.1  sec.      10.    Half  a  mile. 

Art.  130.    4.    (1)  37;  (2)  ^^i^a^;  (3)  Qa^;  (4)   -fa^^;  (5)  iTrafec; 

(6)  l^raB;    (7)  ^';    (8)  '^';    (9)  \-.a^-^a\ 

Art.  131.     3.    5. 

Art.  132.     5.    1154.7  cu.  in.     6.   faHana.     7.   K^r  -  f)a^     8.   2720.3 

cu.  in.  ;  ^^  tan  a. 

2 

Art.133.     4.    f7r(rt2_62)f 


CHAPTEE   XVI. 

Art.  135.      4.    301.6;    ^Trahh.              5.    55|cu.ft. 

6.   faft^cota. 

7.    |(3  7r  +  8)a3.        8.    ^aVi. 

Art.  136.     2.   '^I-     3.    |;   ^.     5.    fTra^.     6.    11  tt. 

7.    fa2. 

Art.  137.      2.    (1)  2  7ra;  (2)   (&)  2{\/2  +  log  (v^  +  l)}a  ; 

(3)4afcos^-cos^V8a;  (4)  «  (/«  -  e"?),  «f  e -IV     3.  i^^^^±«^±^. 
^  ^        V        2  2/'        '  ^  ^2^  ^'2V        ey  a+6 

Art.  138.     2.    (3)  '-^ ;    (4)   (a)  Z  sec «,  in  which  Z  is  the  difference  in 

length  of  the  radii  vectores  to  the  extremities  of  the  arc  ;  (4)  (h)  like  (4)  (a) ; 

(5)  ^  r^2  VlT^^  -  e^^We?  4-  log  h±}^l±Jl\  ■       (6)  a  tan  ^  sec  ^  + 
2L  ^j+Vm^J  2         2 


a  log  tan  [^  +  -") ;  2  a  [sec  ~  +  log  tan  -  ttV 


Art.  139.     5.   47ra2.  6.    ttOk -2)a^.  7.    2 irfta  +  2  Trtft  ^^^Z^. 

e 

8.    (1)  37ra2;    (2)  5  7r2a3,  ^ira'^;    (3)  7r2a3,  ^7ra2.  9.   2  7r2a2&,  ^ir^ah. 

10.    2  7ra2(l_iy       12.    ^7ra3(22  +  37r);   -J— Ta2(7r  +  4). 

V        e)  2V2 

Art.  140.     2.   4(z2.        3.   47ra2.         4.    Surface  =  8af2  6sin-i -—^=:^ 
■      7,2     X  \  y/¥^^' 

—  a  sin-i  — - —  ] ' 

«2  -  b^l 


ANSWERS.  429 

Art.  141.     3.    1341;  91.        4.    4.64.        5.    (1)  2|,  5^;    (2)  f,  1.14,  .94; 

(3)  5^,  9^.  6.  (1)  9.425;  (2)  15.71;  (3)  1.571  &,  1.571a.  7.  — ,  — • 
9.  InK  10.  1.273  a.  12.  1.132  a,  1.5  a2.  13.  |a,  Ja^.  14.  32.704°. 
15.  J  a,  fa.     16.  fa,  fa2.     17.  fa,  fa^.     18.  1.273  a,  2  a^     19.  .6366  a,  ^  a^. 

CHAPTER  XVII. 

Art.  143.     1.    (1)  First  order  at  (1,  1)  ;      (2)  second  order  at  (2,  8). 
2.  ?/  =  5x2-6x  +  3.     3.-1.     4.   ^  =  3a;2-3x+ 1.     5.  1/ =  x2  -  3x  +  3. 

Art.  144.    1.    5.27  and  (-  4,  |)  ;  2.635  and  (-  |,  Jgi).         2.    B  =  145.5  ; 
(-  143,  20^2). 

Art.  148.     1.   The  curvature  of  y=x^  is  one-half  that  of  y  =  6x2— 9x+4. 

2.   — ;  i?=-88.4;  (-87.5,  -12.5). 
125'  '  ^  ^ 

Art.  149.     3.   liP±^;        f2p  +  3x,    - -^\;    -  2p  and    (2p,    0). 

pi  \  ^p'y 

4.    ^^_XMx^+aw!^  (o!,:^^.  Centre  at  l^^^^x%  -^^-=^yA. 
a%^  ab         '  V     «*  b^     ^  J 

5     a^  j>  ^  i^'x^  +  aV)^  ^  (e2x2-a2)l  ^  /a^  +  b\^        «^  +  6%.\ 

^  ^  a4&*  aft         '  V     a*         '  6*     ^  A 

^^>  ^^  (*-.^.  ^^-a- '''  'i'  {-'-'^^  '^)- 

(4)  -  3  (iaxyy  ;  (x  +  3  v^,  1/  +  3  v^).  (5)  3  a  sin  0  cos  0  ;  (a  cos^  « 

3    1 
+  3acos«sin2^,  asin3^+3asin^cos2«).      (6)    (4q  +  9x)^x^.    /^_o^^ 

Qa  \  a 

4y  +  |^y      (7)   -2a;  (a,  -fa).      (8)  ±4asin-;  {a  ■  6  +  sin  ^,  -  y). 

6.    (1)  (£±J}!.    (2)  («*  +  9a;^)^     (3)  csec?-     (4)  |a.     (5)  2acosec3f. 
2  Va  6  a^x  c 

^g^^    (a2sin2  0  +  &2cos2  0)f       ^^^        (a2tan2  0  +  62sec2  0)i  asec2^, 

2  a&  ab 

i.e.  ^• 
a 

Art.  150.    1.(1)  a;  6.    (2)^^-     (3)  |  V2^.     (4)  -  4     (5)  ±  f^- 

V  a  ^  3  r 

(6)  rv/TT^.     (7)   ±<1  +  ^.     (8)   ^a0"-Hn^  +  </>^)l 

Art.  151.     3.    (1)  (ax)^-(6?/)*=(a2  +  62)l.     (2)  (x-\-y)^  -  (x -y)^ 

=  (4  a)i     [Suggestion :  Show  that  a -¥  ^  =  -(--^-Y,  a-3  =  ^(^--]\ 

2\x      a)  2  \x     a) 

and  deduce  therefrom.]     (3)  (x  +  ?/)^  +  (x  —  ?/)  ^  =  2  a^' 


430  INFINITESIMAL   CALCULUS. 


CHAPTER   XVIII. 

Art.  157.  5.  (1)  x2  +  ?/2  =  a2.  (2)  h^yi^  +  a'^y'^  =  a%\  (3)  iay^-\-bxy 
-\-cx^  =  4.ac-  h\  (4)  4  x^/  +  a^  =  0.  (5)  4  i/^  =  27  a'^x.  (6)  [x  -  a)^ 
+  (y  _  5)2  =:  ^2.     6.    (1)  a;2  +  ?/2  =  a2,     (2)  a;2_^2:=^2.     (^s)  (axy  +  (byy^ 

=  (^2-52)1.  7.  (1)  The  lines  a;  ±  ?/ =  0  ;  (2)  27  cy^  =  4  x^  10.  A  parab- 
ola ;  1/2  =  4  ax  if  the  fixed  point  be  (a,  0)  and  the  fixed  line  be  the  jz-axis. 

Art.  158.     3.   4x?/  =  a2.      4.   ^jl  _|_  ^1  =  ^f       5.   x^  +  yt  =  ^^1      , 

Art.  160.  4.  (l)x  =  a,y  =  b.  (2)  x  =  2.  (3)  ?/ +  3  =0,  2x  + 3  =  0. 
(4)2/  +  l=0.  5.  (2)  (2,  I).  (3)  (-1,  -3),  (-1,  -V).  (4)(-i,  -1). 
8.  (1)  x=0,  y=0.     (2)  x=2a.     (3)  y=0.      (4)  x=±a,  2/=±  &.    (5)  ?/=0, 

x  =  a.  (6)  x=0.  0)  y=0.  (8)  ?/=0.  (9)  x=(±2  w  +  1) -,  in  which  w  is 
any  integer. 

Art.  161.     2.    te±«y  =  0.        5.    (1)  ?/ =  x.     (2)  x  +  y  =  1,  x  ~  y  =  1. 

(3)  x=2,  |/+3=0,  2(^-x)^5.       (4)  x=2/±l,  x  +  ?/=±l.      (5)  6y  =  Sx+2. 

Art.  162.  2.  (1)  Lines  parallel  to  the  initial  line  and  at  a  distance 
±  nair  from  it,  n  being  any  integer.  (2)  The  line  perpendicular  to  the  initial 
line,  at  a  distance  a  to  the  left  of  the  pole.  (3)  The  two  lines  which  are 
parallel  to  the  initial  line  and  are  at  a  distance  2  a  from  it.  4.  r  sin  (^— 1)  =1 ; 
r  =  l. 

Art.  165.  3.  (1)  Node  at  origin  ;  slopes  there  are  ±  1.  (2)  Cusp  at 
(—3,  1)  ;  slope  there  is  0.  (3)  Cusp  at  (2,  1) ;  tangent  there  is  parallel  to 
the  ?/-axis.  (4)  Double  point  at  (0,  0)  ;  slopes  of  tangents  there  are  1,  —  |. 
(5)  Cusp  at  (1,  2)  ;  slope  of  tangent  there  is  1.  (6)  A  conjugate  point 
at  (3,  0). 

CHAPTER   XIX. 

Art.  171.     3.    (1)  Convergent.  (2)  Convergent.  (3)  Divergent. 

(4)  Divergent  except^ V.ben  p  >  2.  (5)  Convergent  if  p  >^'2y  4.  (^);^^*<1, 
convergent;  x'>  1,  or  .x,=  1^  divergent.  -(2)  Xbsolutely  convergent  if 
x2  <4,  divetgent  if  x2  =  1,  divergent  if  x2  >  1.  (3)  Absolutely  convergent 
for  all  values  of  x.        (4)  x  <  1,  or  x  =  1,  convergent;   x>l,  divergent. 

(5)  Same  as  in  Ex.  4. 

CHAPTER  XX. 

Art.  176.     5.    (a)  cosx  -  ^sinx  -  —  cosx  +  —  sinx  H ;      (6)  cos^ 

^2„„..x3  2!  3! 


2!  3! 

Art.  177.     4.  e  +  e(x  -  1)  +  ^  (x  -  1)2  + 

Ji  I 


ANS IVERS.  431 

Art.  178.     10.  (1)  1 +  — +  — +  ^^+ •••;        (2)   ?^  +  ^  +  ^+.... 
^  ^  2  !       4  !         6 !  2       12      45 

12.    (l)c  +  x  +  f-|f-?^-?^^?^-f...;(2)log^M^-a) 


2.213.3!  '  '         a  -3      1-2.5      1.2.3-7 

CHAPTER   XXI. 


Art.  186.     2.  2/  Vl  -  x-*  +  x  Vl  -  ^^  =  c.      3.  (y  +  6)"(ic  +  a)"*  =  c. 

Art.  187.     1.  x2  +  i/2  =  cy.     2.  x2(x-2  +  2  y'^)  =  c*.     3.  Xi/2  =  c^(x  +  2  ?/). 
4.  xy(x  -y)  =c. 

Art.  188.     1.  xy  =  c.        2.  x2y  +  3  X  +  2  y2  _  c.         3.  gx  gin  y  +  x^  =  c. 

4.  3  axy  -  y^  =  x^  +  c.        7.  a  log  (x^y)  -  y  =  c.        8.  log  —  =  — • 

y      xy 


Art.  189.     3.   vl  —  x"^  .  y  =  sin-i x  +  c.  4.  y  =  tan x  —  1  +  ce"'*"*. 

b.  y  =  x2(l  +  ce^.    7.  Sy^  =  c(l  -  x^)^  -  1  +  x2.     8.  y^^x^  +  1  +  ce'')  =  1. 

Art.  190.     2.  2/2  =  2  ex  +  c^.  3.  y  =  c  -  [i?2  +  2p  +  2  log  (p  -  1)], 

X  =  c  -  [2p  +  2  log  (p  —  1)].       4.  log  (p  —  x)  =  -^^ H  c,  with  the  given 

p-x 

relation.  5.  (x^  +  y)^  {x^-2y)  +2  x(x^  -Sy)c  =  cK         6.  y  =  ex  +  -. 

7.  y  =  ex  +  a  VH-  e2.        8.  2/2  =  cx2  +  c2.  ^ 

Art.  191.     2.  x''«  +  y2  =  a2;  x2(x*  -  4  2/2)  =  0.  3.   (1)  2/ =  ex +c2, 

x2  +  42/  =  0.  (2)  (2/  +  x-c)2  =  4x2/,  x2/=0.  (3)  (x -y  +  c)^  =  a{_x  ^  y)^, 
X  +  2/  =  0. 

Art.  192.  3.  The  concentric  circles  x2  +  2/^  =  a2,  4,  The  lines  y  =  mx. 
B.  (1)  The  ellipses  y^  +  2  x2  =  e2  ;   (2)  the  hyperbolas  x2  -  ^2  _  ^2  ;  (3)  the 

conies  x2  +  ny^  =  c  ;  (4)  the  curves  y^  —  x^  =  c^ ;  (5)  the  ellipses  x2  + 
2  2/2  =  c2  ;  (6)  the  cardioids  r  =  e(l  +  cos  d)  -,  (7)  the  curves  r**  cos  w^  =  c"  ; 
(8)  the  curves  r^  =  c^  sin  nd  ;  (9)  the  lemniscates  r^  =  c^  sin  2  ^,  whose  axes 
are  inclined  at  an  angle  45°  to  the  axes  of  the  given  system  ;  (10)  the  con- 
focal  and  coaxial  parabolas  r(l  —  cos  ^)  =  2  e  ;  (11)  the  circles  x2  +  2/2-2  Ix 
+  a2  =  0,  in  which  I  is  the  parameter.  10.  The  conies  that  have  the  fixed 
points  for  foci.  11.  The  conies  that  have  the  fixed  points  for  foci.  12.  The 
conies  62a;2  _[.  a^y'2  =  ^262.  13.  The  hyperbola  4  xy  =  a"^.  14.  The  parabola 
(x  -  2/)^  -  2  a(x  +  2/)  +  a2  =  0. 

Art.  193.  3.  (1)  2/=e^(acos3x+6sin3x).  (2)  2/=Cie2^+C2e^+C8e*'. 
(3)  y  =  Cie*^-\-e-^(c2-^Csx).  (4)  2/ =  e2x(ci -f  e2x)  + e*'(e3  cos  5  x+e4  sin  5  x). 
7.    (I)  2/  =  X (a  cos  log  X  4-  6  sin  log  x).  (2)  y  =  x(ci  -\-  c^  log  x). 

(3)  2/  =  x2(ei  +  C2  log  x).  (4)  y  =  CiX'^  +  x(c2  cos  logx  +  C3  sin  log  x). 

9.    2/  =  (5  +  2  x)2{ei(5  +  2  x)v'2  _,.  02(5  +  2  x)-v^2^ 


432  INFINITESIMAL   CALCULUS, 

Art.  194.     4.    (1)  y  =  cie«^  +  C2e-«*.  (2)  e2cx  _|.  2  ccie^-y  =  ci2. 

(3)  t  =^5^  { I  (vers-i  ?^  -  tt^  -  v/aa;  -  a;^  | .     6.   The  circle  of  radius  a. 

6.  (1)  ?/=Cia;+(c'iHl)log(a;-Ci)+C2.  (2)  y  =  Ci  log  x  +  cg.  (3)  2(^-&) 
=:gx-«4.g-(x-a),  (4)  j,^cilog(l-fx)+^x-ia;Hc2.  8.  (1)  y2=a;2_|.Cia;  +  C2. 
(2)  log^  =  cie^+C2e-^.    (3)  {x-cxY=C2{y'^  +  C2).     (4)  ?/  =  logcos(ci-x)  +  e2. 

Page  351.     (l)r=asin^.    (2)  xeJ'=c(l  +  x-|-y),    (3)  c(2?/2+2a:y-a:2)2>/3 

=  (V3+l)x  +  2y^     ^^>j  x2  =  2  cy  +  c^.     (5)  |/  sec  a;  =  log  (sec  x  +  tan  x)  +  c. 

(l_V3)x  +  2?/ 
(6)  3y  =  x2(l  +  a;2)^  +  cx^.  (7)   3x2  +  4xy  +  6?/2  +  5x  +  ?/  =  c. 

(8)   (x  -  2  c)y2  =  c2x.      (9)  ?/(x2  +  1)2  =  tan-J  X  +  c.      (10)  602/3(a;  +  1)2  _ 

10 x6  -f  24 x^  +  15 X*  +  c.  (11)  X  =        ^        {_c-\-  a sin-^jo),  y  =-  ap-\- 

Vl-i)2       

—  (c  +  asin~ip).         (12)  x  +  c  =  alog  (jp  +  Vl  +^^2),  ^  =  a  Vl  +  P^ 

(13)  ^2:^cx2--i^.      (14)  x  =  cx?/  +  c2.      (15)   y  ==8(^924.^3)  +  | log (2p- 1), 

c  +  1  

(16)  ?/(l±cosx)=c.  (17)  ?/2+(x+c)2=a2.  ^2=0^2.  (Ig)  ?/=cx+ V&2+a2c2; 
62x2  -h  aY  =  «^&^-  (19)  9(2/  +  c)2  =  4 x(x  -  3  a)2  ;  X  =  0.  (20)  y  =  Cie«« 
+  C2e-«*  +  C3  sin  (ax  +  «).     (21)  y  =  (cie*  +  C2e-*)  cosx  +  (cge*  +  046-*)  sin  x. 

(22)  2/  =  e2x(-ci  +  C2X)  +  cse-*.  (23)  y  =  CiX -{■  CsX-i.  (24)  2^  =  -^  + 

x^|c2Cos[  — logx)  +  cssin(  —  logxYl.     (25)  2/  =  Ci(x  + a)2  + C2(x+a)3. 

(26)  (cix  4-  C2)2  +  a  =  Ciy\  (27)  3  x  =  2  a^(y^  -  2  c{)  {y^  +  Ci)^  +  Co. 

(28)  y  =  ci  log  X  +  ^  x2  +  C2.     (29)  e-«y  =  CxX  +  C2. 


INDEX. 


[The  numbers  refer  to  pages.} 


Abdank-Abakanowicz,  169. 

Absolute,  constants,  16;  value,  14. 

Acceleration,  109. 

Adiabatic  curves,  87. 

Aldis,  Solid  Geometry,  299. 

Algebra,  Chrystal's,  14,  29,  67,  70,  etc. ; 
ChrystSiVs  Introduction  ^o,  19 ;  Hall 
and  Knight's,  70,  303,  etc. ;  Hall's 
Introduction  to  Graphical,  19. 

Algebraic  equations,  theorems,  98,  99. 

Algebraic  functions,  17,  61,  98. 

Allen,  see  *  Analytic  Geometry.^ 

Amsler's  planimeter,  229. 

Analytic  Geometry,  Ash  ton,  131 ;  Candy, 
5;  Tanner  and  Allen,  19,  131; 
Wentworth,  131. 

Analytical  Society,  45. 

Anti-derivatives,  50,  53, 

Anti-differentials,  50,  170,  171. 

Anti-differentiation,  148,  170. 

Anti-trigonometric  functions,  17. 

Applications:  elimination,  114;  equa- 
tions, 98,  99;  geometrical,  84; 
physical,  84 ;  rates,  91 ;  of  partial 
differentiation,  141 ;  of  integra- 
tion, 192,  etc. ;  of  successive  inte- 
gration, 235,  etc. ;  of  integration 
in  series,  313 ;  of  differentiation  in 
series,  313;  of  Taylor's  theorem, 
322,  331,  332. 

Approximate  integration,  223. 

Approximations :  to  areas  and  integrals, 
157,  223,  316;  to  values  of  func- 
tions, 49;  to  small  errors  and 
corrections,  96,  138. 

Arbitrary  constants,  16. 

Arbogaste,  42. 

Arc:  derivative,  102,  103;  length,  245, 
248 ;    Huyghen's    approximation. 


Archimedes,  see  '  Spiral.' 

Area :  approximation  to,  223, 225 ;  deriv- 
ative, differential,  99,  101 ;  me- 
chanical measurement,  228,  229; 
of  curves,  192,  242,  244;  of  a 
closed  curve,  198,245;  of  surfaces 
of  revolution,  249;  of  other  sur- 
faces, 253 ;  precautions  in  finding, 
198 ;  sign  of,  197,  245 ;  swept  over 
by  a  moving  line,  245. 

Argument  of  function,  15. 

Ashtou,  see  'Analytic  Geometry.' 

Astroid,  see  '  Examples.' 

Asymptotes,  286;  circular,  292;  curvi- 
linear, 291 ;  oblique,  290 ;  parallel 
to  axes,  288;  polar,  292;  various 
methods  of  finding,  291. 

Asymptotic  circle,  292. 

Average  value,  259. 

Beman,  Famous  Problems,  13. 

Bernoulli,  150, 

Binomial  Theorem,  322. 

Bitterli,  169. 

Borel,  divergent  series,  305. 

Burmaun,  18. 

Byerly,  see  '  Calculus.' 

Cajori,  History  of  Mathematics,  42,  46, 
149,  204,  222. 

Calculation  of  small  corrections,  96. 

Calculus,  1;  differential,  11,  39,  149; 
integral,  11,  39,  50,  149;  funda- 
mental theorems,  36;  invention,  1, 
149;  notions  of,  11. 
references  to  works  on  :  Byerly, 
Diff\  34,  etc.;  Byerly,  Problems, 
111,  etc.;  Echols,  41,  etc.;  Ed- 
wards, Integral,  213,  etc.;  Ed- 
wards, Treatise,  129,  etc. ;  Gibson, 
29,  etc. ;  Harnack,  20,  etc. ;  Lamb, 


433 


434 


INFINITESIMAL   CALCULUS. 


29,  etc. ;  McMahon  and  Snyder, 
Biff.,  33,  etc. ;  Murray,  Integral, 
163,  etc.;  Perry,  12,  381,  etc.; 
Smith,  W.  B.,  135,  222;  Snyder 
and  Hutchinson,  156,  etc. ;  Taylor, 
129,  etc.;  Todhunter,  Diff.,  70, 
etc. ;  Integral,  163 ;  Williamson, 
Diff.,  70,  etc.;  Integral,  163,  etc.; 
Young  and  Linebarger,  381. 

Candy,  see  'Analytic  Geometry.' 

Cardioid,  see  '  Examples.' 

Catenary,  see  '  Examples.' 

Cauchy,  304;  form  of  remainder,  327. 

Centre  of  curvature,  269,  270;  of  mass, 
373. 

Change  of  variable,  in  differentiation, 
145;  in  integration,  175. 

Chrystal,  see  ^Algebra.' 

Circle,  curvature  of,  267 ;  of  curvature, 
268 ;  osculating,  264 ;  see  *  Exam- 
ples.' 

Circular  asymptotes,  292. 

functions  and  exponential  functions, 
327. 

Cissoid,  see  '  Examples.' 

Clairaut's  equation,  339. 

Commutative  property  of  derivatives, 
133, 140. 

Comparison  test  for  convergence,  307. 

Complete  differential,  136. 

Compound  interest  law,  70. 

Computation  of  tt,  313,  314. 

Concavity,  260. 

Condition  for  total  differential,  141. 

Conjugate  points,  295. 

Conoids,  241. 

Constant:  absolute,  16;  arbitrary,  16; 
elimination  of,  114;  of  integra- 
tion, 160,  162,  166,  335. 

Contact:  of  curves,  261,  280,  282;  of 
circle,  263;  of  straight  line,  262. 

Continuity,  continuous  function,  see 
'Function.' 

Convergence :  kinds  of,  304,  305,  311 ; 
interval  of,  306;  tests  for,  307, 
308;  see  '  Series,'  '  Infinite  Series.' 

Convexity,  260. 

Corrections,  96. 

Cos  X,  derivative  of,  73;  expansion  for, 
322,  325. 

Criterion  of  integrability,  188. 

Critical  point,  critical  value,  118, 120. 

Crossing  of  curves,  263,  332. 

Cubical  parabola,  see  '  Examples.' 


Curvature:  average,  266;  at  a  point, 
266,  267;  total,  266;  centre  of, 
269,  270;  of  a  circle,  267;  circle 
of,  268 ;  radius  of,  268,  271. 

Curves:  area  of,  192,  242,  244;  asymp- 
totes, 286;  contact  of,  261,  280, 
282;  derived,  44;  differential,  44; 
envelope,  277;  equations  derived, 
203;  evolute,  272;  family,  277; 
integral,  168,  169;  involutes,  276; 
length,  245,  248 ;  locus  of  ultimate 
intersections,  278 ;  Loria's  Special 
Plane,  2^;  parallel,  276;  see  'Ex- 
amples.' 

Curve  tracing,  298. 

Curvilinear  asymptotes,  291. 

Cusps,  280,  293,  294,  296,. 297. 

Cycloid,  see  'Examples.' 

Decreasing  functions,  116. 

Definite  integral,  sec  '  Integral.' 

De  Moivre's  theorem,  328. 

Density,  372. 

Derivation  of  equation  of  curves,  203. 

Derivative :  definition,  38 ;  notation,  41 ; 
general  meaning,  4(5;  geometric 
meaning,  43;  physical  meaning, 
45;  progi'essive,  regressive,  93. 

Derivatives:  of  sum,  product,  quotient, 
51,  53-57;  of  a  constant,  52;  of 
elementary  functions,  61-80 ;  of  a 
function  of  a  function,  59;  of  im- 
plicit functions,  80,  1.39  ;  of  in- 
verse functions,  61 ;  special  case, 
60;  geometric,  99-106;  successive, 
107,  112;  meanings  of  second,  108, 
109. 

Derivatives,  partial,  81,  130,  131 ;  com- 
mutative property  of,  133,  140; 
geometrical  representation,  132; 
successive,  133. 

Derivatives,  total,  136;  successive,  143. 

Derived,  curves,  44;  functions,  40,  41. 

Descartes,  149. 

Differentiable,  41. 

Differential  calculus,  see  '  Calculus.' 

Differential  coefficient,  see  '  Derivative.' 

Differential,  differentials,  47,  49;  com- 
plete, 136;  exact,  142;  geometric, 
99-106;  infinitesimal,  155 ;  partial, 
136;  successive,  112;  total,  136, 
137 ;  condition  for  total,  141 ;  in- 
tegration of  total,  188. 

Differential  equations,  114,  334 ;  classifi- 


INDEX. 


435 


cation,  334;  Clairaut's,339;  exact, 
336;  homogeneous,  336;  linear, 
337, 34(i,  348 ;  order,  334 ;  ordinary, 
334;  partial,  334;  second  order, 
349;  solutions,  114,  335,  340 ;  refer- 
ences to  text-books,  115,  351,  etc. 

Differentiation,  39, 170;  general  results, 
51 ;  logarithmic,  68 ;  of  series,  312 ; 
successive,  107;  see  'Derivative,' 
'  Derivatives.' 

Discontinuity,  discontinuous  functions, 
see  '  Functions.' 

Divergent  series,  see  '  Series.' 

Double  points,  280,  293,  294. 

Doubly  periodic  functions,  221. 

Durand's  rule,  227. 

Echols,  see  '  Calculus.' 
Edwards,  see  '  Calculus.' 
Elementary  integrals,  172, 180. 
Elimination  of  constants,  114. 
Ellipse,  see  'Examples.' 
Ellipsoid,  235. 
Elliptic  functions,  158,  221. 
integrals,  158,  221,  317. 
End-values,  155. 

Envelopes,  contact  property,  280,  282; 
definition,   278;    derivation,  '281, 
284. 
Equations,   derivation  of,  203;  graphi- 
cal   representation,   19,    20,    130 ; 
roots  of,  98,  99. 
Equiangular  spiral,  see  '  Examples.' 
Errors,  small,  9(j,  138;  relative,  96. 
Euler,  142,  314,  328 ;  theorem  on  homo- 
geneous functions,  142. 
Evolute,  definitions,  272,  275,  283. 

properties  of,  273. 
Evolute  of  the  ellipse,  see  '  Examples.' 
Exact  differential,  142. 

equations,  336. 
Examples  concerning: 
adiabatic  curves,  87. 
astroid  (or  hypocycloid),  86, 102,  198, 

203,  251,  270,  273,  .345,  365. 
cardioid,  91,  101,  244,  249,   252,  271, 
345,  365,  .376,  383,  390,  .392,  393,  396. 
catenary,  201,  248,  25.3,  366,  383. 
circle,  86,  194,  202,  244,  249,  252,  271, 

344,  375,  376,  378,  391,  393,  399. 
cissoid,  290,  393. 
cubical    parabola,  92,  101,  102,  158, 

166,  195,  198,  201,  270. 
cycloid,  87,  248,  252,  270,  273,  366. 


ellipse,  86,  106,  200, 203,  248,  257,  276, 

290, 374,  385,  390,  391,  393,  397,  399, 

400. 
evolute  of  the  ellipse,  273,  390. 
exponential  curve,  86,  392. 
folium  of  Descartes,  87,  244,  290. 
harmonic  curve,  398. 
hyperbola,  87,  92,  270,  271,  273,  290, 

291,  345,  383,  388, 390,  393. 
hypocycJoid,  see  '  Astroid.' 
lemniscate,  244,  271,  345,  383,  393. 
limaQon,  398. 
parabola,  86,  87,  92,  102, 104, 152, 159, 

166, 195, 196,  198,  234,  249,  257,  270, 

271,  273,  276,  283,  284, 290,  345,  366, 

376,  383,  388,  391,  393. 
probability  curve,  290. 
semi-cubical    parabola,   86,  87,   159, 

198,  270,  366,  391. 
sinusoid,  86,  159. 
tractrix,  3(Ki,  392. 
the  witch,  87,  271,  290. 
Spirals : 

archimedes',  01,  101,  103,  249,  271. 

equiangular  (or  logarithmic),  91, 
^14,  249,  271,  366,  390. 

general,  91,  271. 

hyperbolic  (or  reciprocal),  91,  244, 

logarithmic,  see  '  Equiangular.' 

parabolic  (or  lituus),  91. 

reciprocal,  see  '  Hyperbolic' 
Expansion  of : 
cos  X,  322,  325. 
log  (l  +  x),  logarithmic   series,  315, 

321. 
sin  X,  322,  325. 
sin-ia;,314. 

e',  exponential  series.  .326. 
tan"i  X,  Gregory's  series,  228,  313. 
Expansion  of  functions : 

by  algebraic  methods,  326. 
by  differentiation,  228,  315. 
by  integration,  228,  313. 
by  Maclaurin's  series,  325. 
by  Taylor's  series,  321. 
Explicit  function,  17. 
Exponential  curve,  see  '  Examples.' 
function,  17;  expansion  of,  326;  and 

trigonometric,  relations  between, 

327. 

Family  of  curves.  377. 
Fermat,  122,  149,  247. 
Fluent,  fluxion,  45. 


436 


INFINITESIMAL   CALCULUS, 


Folium  of  Descartes,  see  '  Examples..' 

Forms,  indeterminate,  367. 

Formulas  of  reduction,  213,  218. 

Fourier,  155. 

Fractions,  rational,  integration  of,  184. 

Frost,  Curve  Tracing,  291,  293,  299. 

Function,  15;  algebraic,  17,  61,  221;  cir- 
cular, 221 ;  classification,  17 ;  con- 
tinuous, 25,  28,  41,  130;  derived, 
40,  41 ;  discontinuous,  25,  27 ; 
elliptic,  158,  221;  explicit,  17; 
exponential,  17,  70;  graphical 
representation,  19,  20,  130;  homo- 
geneous, Euler's  theorem  on,  142; 
hyperbolic,  183,221,  353;  implicit, 
17,  80;  increasing  and  decreasing, 
116;  inverse,  18,  76;  irrational, 
206;  logarithmic,  17,  67;  march 
of  a,  123 ;  maximum  and  minimum 
values  of,  117 ;  notation  for,  17, 
18;  of  a  function,  59,  60;  of  two 
variables,  130 ;  periodic,  221 ;  tran- 
scendental, 17 ;  trigonometric  and 
anti-trigonometric,  17,  71,  215; 
turning  values  of,  117;  variation 
of,  116. 

Gauss,  304. 

General  integral,  see  'Integral.' 

spiral,  see  '  Examples.' 
Geometrical  interpretation,   a  certain, 

215. 
Geometrical  representation  of : 

derivatives,  ordinary,  43. 

derivatives,  partial,  132. 

functions  of  one  variable,  19,  20. 

functions  of  two  variables,  20,  130. 

function  of  a  function,  60. 

integrals,  definite,  163. 

integrals,  indefinite,  166. 

total  differential,  137. 
Geometric  derivatives  and  differentials, 

99-106. 
Geometry,  Famous  Problems  in,  13. 
Gibson,  see  '  Calculus.' 
Glaisher,  Elliptic  Functions,  222. 
Graphical  Alr/ebra,  Hall,  19. 
Graph  of  a  function,  19. 
Graphs,  sketching  of,  123. 
Gregory,  305,  313. 
Gregory's  series,  313. 
Gyration,  radius  of,  377. 

Hardy,  infinitesimals,  38. 

Harkuess  and  Morley,  Analytic  Func- 


tions, 15,  29,  303 ;  Theory  of  Func- 
tions, 41.  J 

Harmonic  curve,  398. 

Harmonic  motion,  83,  110. 

Harmonic  series,  304. 

Harnack,  see  '  Calculus.' 

Hele  Shaw,  Mechanical  Integrators, 
229. 

Henrici,  Report  on  Planimeters,  229. 

Herschel,  18,  46. 

Hobson,  Trigonometry,  303,  314,  363. 

Homogeneous,    differential    equations, 
336. 
functions,  Euler's  theorem,  142. 
linear  equation,  348. 

Horner,  Horner's  process,  324,  333. 

Hutchinson,  see  '  Calculus.' 

Huyhen's  rule  for  circular  arcs,  326. 

Hyperbola,  see  '  Examples.' 

Hyperbolic  functions,  183,  221,  328,  353. 
spiral,  see  '  Examples.' 

Hypocycloid,  see  '  Examples.' 

Implicit  functions,  differentiation,  8G> 
139. 

Increasing  function,  116. 

Increment,  notation  for,  4. 

Indefinite  integral,  see  '  Integral.' 

Indeterminate  forms,  25)6,  367. 

Inertia,  centre  of,  373. 
moment  of,  377. 

Infinite  numbers,  14,  30,  31. 
orders  of,  31. 

Infinite  series,  300;  algebraic  proper- 
ties, 304;  differentiation  of,  302, 
312;  general  theorems,  305;  inte- 
gration in,  227,  316;  integration 
of,  302,  310;  limiting  value  of, 
301;  questions  concerning,  301; 
Osgood,  article  and  pamphlet, 
303,  306,  307;  remainder,  306; 
study  of,  303. 
See  '  Series.' 

Infinitesimal,  1,  48,  49. 

Infinitesimal  differential,  155. 

Infinitesimals,  30;  Hardy,  38;  orders, 
31;  summation,  150;  theorems, 
33,  35. 

Inflexion,  points  of,  118,  127,  261. 

Inflexional  tangent,  129. 

Integral  curves,  168,  169. 

Integral,  definite,  approximation,  223, 
316;  definition,  representation  of, 
properties,  164-158, 163,  164. 


INDEX. 


437 


Integral :  double,  230 ;  element  of,  155 ; 
elementary,  172, 180 ;  elliptic,  158, 
221,  248,  317 ;  general,  162. 

Integral,  indefinite,  162;  representa- 
tion of,  166. 

Integral:     multiple,    231;     particular, 
162;  precautions  in  finding,  198; 
triple,  230. 
See  '  Calculus.' 

Integrand,  150. 

Integraph,  169,  228,  229. 

Integrating  factors,  336. 

Integration,  148, 170 ;  as  summation,  154, 
170 ;  as  inverse  of  differentiation, 
160;  constant  of,  160,  162,  166; 
general  theorems  in ,  173 ;  succes- 
sive, 230,  232. 

Integration :  by  parts,  177 ;  by  substitu- 
tion, 175,  183,  215 ;  by  infinite  se- 
ries, 227,  316;  by  mechanical 
devices,  228. 

Integration  of :  infinite  series,  302,  310; 
irrational  functions,  206 ;  rational 
fractions,  184;   total  differential, 
188 ;  trigonometric  functions,  215. 
See  '  Applications.' 

Integrators,  228,  229. 

Intrinsic  equation,  249,  363. 

Invention  of  the  calculus,  1,  149. 

Inverse  functions,  18,  61,  76. 

Involutes,  276. 

Irrational  functions,  integration,  206. 

Isolated  points,  295,  296. 

Jacobi,  133. 

Kepler,  122. 
Klein,  13,  67. 

Lagrange,  42,  149,  326. 

Lagrange's  form  of  remainder,  323. 

Lamb,  see  '  Calculus.' 

Laplace,  149. 

Legend  re,  317. 

Leibnitz,  1,  42,  45,  149,  150,  282,  313. 
theorem  on  derivative    of    product, 
113. 

Lemniscate,  see  'Examples.' 

Lengths  of  curves,  245,  248. 

Lima^on,  see  '  Examples.' 

Limits,  limiting  value,  20,  23,  42; 
Hardy,  38;  theorems,  35;  in  inte- 
gration, 155;  of  a  series,  301. 

Linear  differential   equations:  of  first 


order,  337;  with  constant  coeffi- 
cients, 346 ;  honwJgeneous,  348. 
Linebarger,  see  '  Calci^lus.' 
Lituus,  see  '  Examples.' 
Locus  of  ultimate  intersections,  278. 
Logarithmic,  differentiation,  68. 

function.  17,  67. 

series,  315,  321. 

spiral,  see  '  Examples.' 
Loria,  Special  Plane  Curves,  299. 

Machin,  314. 
Maclaurin,  327. 

theorem  and  series,  324,  328. 
Magnitude,  orders  of,  31. 
Mass,  centre  of,  372. 
Mathews,  G.  B.,  305. 
Maxima  and  minima,  116;  by  calculus, 
117-122;   by  other  methods,  122; 
of  functions  of  several  variables, 
122 ;  practical  problems,  123. 
McMahon,  proof,  142. 

See 'Calculus.' 
Mean  square  value,  259. 
Mean  values,  255. 
Mean  value  theorems : 

differentiation,  84,  94,  95,  318. 

integration,  165,  255. 
Mechanical  integrators,  228. 
Mechanics,  372. 

Mellor,  Higher  Mathematics,  381. 
Mercator,  315. 
Minima,  see  '  Maxima.' 
Moment  of  inertia,  377. 
Morley,  see  '  Harkness.' 
Motion,  simple  harmonic,  83,  110. 
Muir,  on  notation,  133. 
Multiple,  angles  in  integration,  217. 

integrals,  231. 

points,  293,  296. 

roots,  98. 

Neil,  246. 

Newton,  1,45,  149,  314. 

Nodes,  294. 

Normal,  rectangular,  84. 
polar,  89. 

Notation  for:  absolute  value,  14;  de- 
rivatives, 41, 46, 107 ;  differentials, 
47;  functions,  17;  increment,  4; 
infinite  numbers,  15;  integration, 
149,  162,  163,  233;  inverse  func- 
tions, 18,  61 ;  limits,  24 ;  partial 
derivatives,  81, 131, 133, 137 ;  sum- 
mation, 155. 


438 


INFINITESIMAL   CALCULUS, 


1 


Notation,  remark  on,  42. 
Numbers,  13;  algebraic,  13;   finite,  in- 
finite, infinitesimal,  14,30;    tran- 
scendental, 13,  67. 
e  and  tt,  67,  328. 
graphical  representation,  13,  14. 

Oblique  axes,  193. 
Order  of,  contact,  261. 

derivative,  differential,  108,  333. 

differential  equation,  334. 

infinite,  31. 

infinitesimal,  31,  333. 

magnitude,  31. 
Orthogonal  trajectories,  341,  343. 
Oscillatory  series,  304. 
Osculating  circles,  264,  271. 
Osgood,  W.  F.,  article,  311,  313. 

pamphlet,  303,  etc. 

Parabola,  see  '  Examples.' 
Parabolic  rule,  225. 

spiral,  see  '  Examples.' 
Parallel  curves,  276. 
Parameter,  277. 

Partial  derivative,  see  '  Derivative.' 
Partial  fractions,  184. 
Particular  integral,  see  '  Integral.' 
Pendulum  time  of  oscillation,  317. 
Periodic  functions,  221. 
Perry,  on  notation,  133. 

See  '  Calculus.' 
Picard,  156. 
Planimeters,  228,  229. 

Henrici,  Report  on,  229. 
Points,   see   'Critical,'    'Double,'    'Iso- 
lated,' '  Multiple,'  '  Salient,'  '  Sin- 
gular," Stop,'  'Triple,'  'Turning.' 
Power  series,  307,  310,  312,  313. 
Precautions  in  integration,  198. 
Probabilities,  326. 
Probability  curve,  see  'Examples.' 
Progressive  derivative,  93. 

Radius  of  curvature,  268,  271. 

of  gyration,  377. 
Rate  of  change,  11,  45,  46,  47. 

variation,  134. 
Rational  fraction,  integration,  184. 
Reciprocal  spiral,  see  '  Examples.' 
Rectification  of  curves,  246. 
Reduction  formulas,  213,  218. 
Regressive  derivative,  93. 
Remainder  after  n  terms,  306. 


Remainders  in  Taylor's  and  Maclaurin's 

series,  320,  323,  327. 
Ring,  202. 
Rolle,  99. 

Rolle's  theorem,  84,  93,  95,  99,  319. 
Roots  of  equations,  98,  99. 
Rouche  et  Comberousse,  246. 
Rules  for  approximate  integration,  223, 

225,  227. 

Salient  points,  295. 

Schlomilch-Roche's  form  of  remainder, 
327. 

Second  derivative : 

geometrical  meaning,  108. 
physical  meaning,  109. 

Semi-cubical  parabola,  246. 
See  '  Examples.' 

Series,  70;  absolutely  convergent,  305; 
conditionally  convergent,  305 ; 
convergent,  304;  divergent,  304, 
305;  harmonic,  304;  oscillatory, 
304;  uniformly  convergent,  311, 
312. 
See  '  Convergence,' '  Expansion,' '  In- 
finite Series,'  'Power  Series.' 

Serret,  320. 

Sign  of  area,  197,  245. 

Simpson,  Simpson's  rule,  225. 

Sin  X,   sin-ix,    expansions,    314,    322, 
325. 

Singly  periodic  functions,  221. 

Singular  points,  293,  295. 

Singular  solution,  340. 

Sinusoid,  see  '  Examples.' 

Slope,  5,  6,  11,  84,88,  129. 

Slopes,  curve  of,  44. 

Smith,  C,  Solid  Geometry,  253,  299. 

Smith,  D.  E.,  Famous  Problems,  13. 

Smith,  W.  B.,   Infinitesimal  Analysis, 
135,  2*22,  354.' 

Snyder,  see  'Calculus.' 

Solution,  see  '  Differential  Equation.' 

Speed,  2,  3,  4. 

Sphere,  surface,  252,  254. 
volume,  203,  237,  238. 

Spiral,  see  '  Examples.' 

Stationary  tangent,  1*29. 

Stirling,  327. 

Stop  points,  295. 

Subnormal,  rectangular,  84. 
polar,  89. 

Substitutions  in  integration,   175,  207, 
215. 


INDEX. 


439 


Subtangent,  rectangular,  84. 

polar,  89. 
Successive  differentiation,  107. 

derivatives,  107,  112. 

differentials,  112. 

integration,  230,  232. 

of  a  product,  113. 

total  derivatives,  143. 
Summation,  examples,  150. 

integration  as,  154. 
Surfaces,  areas  of,  249,  253. 

volumes,  199,  235,  238,  240. 

Tangent,  5;  inflexional,  129;  length,  84, 

89;  stationary,  129. 
Tanner,  see  ^Analytic  Geometry.^ 
Taylor,  F.  G.,  see  '  Calculus.' 
Taylor's  theorem  and  series : 

applications :  to  algebra,  332 ;  to  cal- 
culation, 49,  137,  322,  323,  324;  to 
contact  of  curves,  332 ;  to  maxima 
and  minima,  331. 
approximations  by,  49,  137,  322. 
expansions  by,  321-324. 
for  functions  of  one  variable,  49,  96, 

318,  319,  323,  328,  330. 
for  functions  of  several  variables,  327. 
forms  of,  320,  321,323. 
historical  note,  326. 
Test-ratio,  308. 
Time-rate  of  change,  45,  135. 
Todhunter,  see  '  Calculus.' 
Total  derivative,  136,  143. 
differential,  136. 
rate  of  variation,  134. 
Tractrix,  see  'Examples.' 
Trajectories,  orthogonal,  341,  343. 


Transcendental  functions,  17. 

numbers,  13. 
Trapezoidal  rule,  223. 
Trigonometric  functions,  direct  and  in- 
verse, 17,  76. 

differentiation  of,  71-80. 

integration  of,  215. 

substitutions  by,  207. 
Trigonometry,  Hobson,  303,  314,  363. 

Murray,  76,  etc. 
Triple  points,  294. 
Turning  points,  values,  117. 

Undulation,  poiuts  of,  128. 
Uniform  convergence,  311,  312. 

Value,  see  *  Average,'  '  Limits,'  '  Maxi- 
mum,'  '  Mean,'    '  Mean    Square,' 
'  Turning.' 
Value  of  TT,  computation  of,  313,  314. 
Van  Vleck,  E.  B.,  305. 
Variable,  dependent,  independent,  13, 15. 

change  of,  145. 
Variation  of  functions,  116. 

total  rate  of,  134. 
Velocity,  92. 

Volumes,  methods  of  finding,  199,  235, 
238,  240. 

AVallis,  149,  246. 

Wentworth,  see  'Analytic  Geometry.' 

Whittaker,  Modern  Analysis,  15, 29,  304, 

309. 
Williamson,  see  '  Calculus.' 
Witch  of  Agnesi,  see  '  Examples.' 
Wren,  247. 

Young,  see  '  Calculus.' 


7 


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